Okay, I'm not familiar with Beilinson's paper, but here's my take on this. First let's recall the two definitions. I will denote the triangulated category of motives over a field $k$ by $DM(k)$ (for any of the equivalent definitions that are available); I am taking $\mathbb{Q}$-coefficients and looking only at compact objects, although I'm not sure the last is necessary. Also, I am not good with homological notation, so I will use cohomological notation all along, beware ! For example, for me $X[1]$ will mean "$X$ put in degree $-1$" and the motive of $\mathbb{G}_m$ will be $\mathbb{Q}\oplus\mathbb{Q}(-1)[-1]$ (where $\mathbb{Q}$ is the unit for the tensor product, i.e., the motive of $Spec(k)$). Sorry, but I'm too afraid to make a mistake if I try to translate.
So, first here is Hanamura's definition of the $t$-structure. He assumes that all the Grothendieck standard conjectures, Murre's conjecture and the vanishing conjectures are true, and this implies in particular that any realization functor $H:DM(k)\longrightarrow D^b(F)$ (where $F$ is an appropriate field of coefficients) is faithful. He defines a $t$-structure, say $({}^H D^{\leq 0},{}^H D^{\geq 0})$ on $DM(k)$ by taking the inverse image by $H$ of the usual $t$-structure on $D^b(F)$. Of course, you have to prove that it is indeed a $t$-structure (and independent of the realization functor), and he does this. He calls the heart the category of mixed motives over $k$, say $MM(k)$. If $X$ is a variety over $k$, we can associate to it a (cohomological) motive $M(X)$ in $DM(k)$, and I will denote by $H^k_M(X)\in MM(X)$ the cohomology objects of $M(X)$ for Hanamura's $t$-structure. Hanamura also proves that every mixed motive has a weight filtration, that is a filtration whose graded parts are pure motives, and he proves that pure motives are semi-simple and that all irreducible pure motives are direct factors of motives of the form $H_M^k(X)(a)$, where $X$ is a smooth projective variety.
Now to Voedvodsky's definition. I have tried to understand it, then rewrite it in cohomological notation, so directions of maps and shifts may have changed, but I think I still got the spirit of it. What he does is something like this : Define a full subcategory ${}^VD^{\geq 0}$ of $DM(X)$ by the condition that an object $M$ is in it if and only if, for every affine scheme $f:U\rightarrow Spec(k)$ that is purely of dimension $n$, for every $m>n$ and every $a\geq 0$, $Hom_{DM(k)}(f_*\mathbb{Q}_U(a)[m],M)=0$. I would like to make a few remarks. First, what I denoted by $f_*\mathbb{Q}_U$ is just my $M(U)$ of the preceding paragraph, but I wrote it like this because it will make the generalization to a general base scheme $S$ more clear; my notation makes sense if I allow myself to remember that we now have categories of motives over a very general base and the 6 operations on them (and if I say that $\mathbb{Q}_U$ is the unit motive in the category of motives over $U$). Second remark, I added twists whereas Voedvodsky's definition doesn't have any. The reason I did this is because Voedvodsky makes a definition only for effective motives, and I didn't see how to make ${}^VD^{\geq 0}$ stable by $(1)$ unless I added it in the definition (but maybe it is not necessary). Third remark, remember, cohomological notation, and for me passing from effective to general motives means inverting $\mathbb{Q}(-1)$, not $\mathbb{Q}(1)$ (in my world, $\mathbb{Q}(-1)$ is effective).
Ah, yes, and then Voedvodsky defines ${}^VD^{\leq 0}$ as the left orthogonal of ${}^V D^{\geq 1}:={}^VD^{\geq 0}[-1]$.
Anyway, what is the motivation for Voedvodsky's definition ? Here are a few principles. First, motivic $t$-structure on motives over a base $S$ should correspond (by the realization functors) to the (selfdual) perverse $t$-structure on complexes of sheaves over $S$. Second, if $f$ is an affine map of schemes, then ${f_*}$ is right $t$-exact for the perverse $t$-structures. Third, for any scheme $U$, the constant sheaves over $U$ are concentrated in perverse cohomology degree $\leq dim(U)$. So, if I come back to my situation above : $f:U\longrightarrow Spec(k)$ is an affine variety over $k$, purely of dimension $n$, $a\geq 0$, $m>n$, then $\mathbb{Q}_U(a)[m]$ should be concentrated in degree $<0$ for the motivic $t$-structure on the category of motives over $U$, and so $f_*\mathbb{Q}_U(a)[m]$ should be concentrated in degree $<0$ for the motivic $t$-structure on $DM(k)$, and it should be left orthogonal to elements that are concentrated in degree $\geq 0$. What Voedvodsky says is that this is enough to characterize the elements concentrated in degree $\geq 0$.
From this, the natural generalization of Voedvodsky's definition to a general base scheme $S$ is obvious : replace affine schemes $U\longrightarrow Spec(k)$ by affine maps $U\longrightarrow S$ (or maps $U\longrightarrow S$ such that $U$ is affine, I don't think it will make a difference).
So, are the two $t$-structures the same ? I think so. A first obvious observation is that ${}^HD^{\geq 0}\subset{}^VD^{\geq 0}$, that is, every object of ${}^HD^{\geq 0}$ is right orthogonal to motives ${{f_*}\mathbb{Q}(a)[m]}$ as above. This follows from the faithfulness of the realization functor and the fact that this would be true in the usual categories of sheaves (see the remarks above). We also know that ${}^HD^{\geq 0}$ is the right orthogonal of ${}^HD^{\leq -1}$, by the definition of a $t$-structure. So, what we have to see is that ${}^VD^{\geq 0}$ is right orthogonal to ${}^HD^{\leq -1}$, that is, that a motive that is right orthogonal to every $f_*\mathbb{Q}_U(a)[m]$ as above is right orthogonal to the whole ${}^HD^{\leq -1}$.
Here is one way to do this : Let $C$ be the smallest full additive subcategory of ${}^HD^{\leq -1}$ that is stable by isomorphism, direct summand, extension and contains all the objects of the form $f_*\mathbb{Q}(a)[m]$ as above. It is enough to show that $C={}^HD^{\leq -1}$. Noting that ${}^HD^{\leq -1}$ is generated (in the same way : direct sumands, isomorphisms, extensions) by objects of the form $H^k_M(X)(b)[l]$, for $X$ a smooth projective variety, $l>k$ and $b\in\mathbb{Z}$, I think that this is an easy exercise, playing with hyperplane sections of smooth projective varieties. (I had a bit a trouble with the fact that $C$ is stable by Tate twists. We know that $M(U)(-1)$ is a direct factor of $M(U\times\mathbb{G}_m)[1]$, so stability by $(-1)$ is not a problem. But I couldn't show stability by $(1)$ unless I put it in the definition.)
edit: corrected dollar sign
As in Birdman's comment, the motivic fundamental group is unifying the notion of monodromy action on the fibers of local systems of "geometric origin."
To explain this, let us start with the case of a field $K$. We have a semisimple $\mathbb{Q}$-linear Tannakian category $\operatorname{Mot}_K$ of (pure) motives over $K$ for which fiber functors are cohomology theories, i.e., it makes sense to have an $L$-valued fiber functor for a field $L$, and this is the same as a Weil cohomology theory for smooth proper $K$-varieties with values in $L$-vector spaces. A motivic Galois group, to my understanding, is attached to a cohomology theory/fiber functor $F$ of $\operatorname{Mot}_K$.
Then the motivic Galois group is the associated group scheme/$L$ whose representations are given by the category $\operatorname{Mot}_{K}\underset{\mathbb{Q}}{\otimes}L$, i.e., it is the group scheme of automorphisms of the fiber functor $F$. So it is "the group which acts on $F$-cohomology of (smooth projective) varieties." Since this category is semi-simple, the motivic Galois group is pro-reductive. E.g., the absolute Galois group (considered as a discrete group scheme) of $K$ acts on $\ell$-adic cohomology, so there is a homomorphism from $\operatorname{Gal}(K)$ to the motivic Galois group of $K$ corresponding to the fiber functor defined by $\ell$-adic cohomology.
For, say, a smooth variety $X$ over $K$, there should be a category of "motivic sheaves" on $X$, or rather, a semi-simple category of pure motivic sheaves contained in an Artinian category of mixed motivic sheaves. You should have e.g. an $\ell$-adic" fiber functor from the mixed category to $\ell$-adic perverse sheaves on $X$ which sends pure guys to (cohomologically shifted) lisse sheaves (alias local systems). E.g., if $K=\mathbb{F}_q$, then this is the category of pure (resp. mixed) perverse sheaves on $X$. If $K=\mathbb{C}$, this should be a full subcategory of pure (resp. mixed) polarizable Hodge modules on $X$. For any smooth proper (resp. just any) map $f:Y\to X$, there should an object in the category of pure (resp. mixed) motivic sheaves on $X$ corresponding to push-forward of the structure sheaf on $Y$.
The motivic fundamental group act on the ``fibers" of pure motivic sheaves on $X$. I.e., for a $K$-point of $X$, you should get a functor to the category of $K$-motives. This is a motivic incarnation of taking the fiber of a local system. Then given our cohomology theory $F$, we obtain a functor from pure motivic sheaves on $X$ to $L$-vector spaces, and the automorphisms of this functor will be the $F$-realization of the motivic Galois group of $X$.
Best Answer
Classically, Grothendieck's motives are only the pure motives, meaning abelian-ish things which capture the (Weil-cohomology-style) $H^i$ of smooth, projective varieties. To see the relationship with motivic cohomology, one should extend the notion of motive so that non-pure (i.e. "mixed") motives are allowed, these mixed motives being abelian-ish things which capture the $H^i$ of arbitrary varieties. The main novelty with mixed motives is that the (conjectural) abelian category of them is not semi-simple -- in fact every mixed motive should be a (generally non-trivial) iterated extension of pure motives, these extensions essentially coming from compactification and resolution of singularities, as in the story of mixed Hodge structures.
Then once one thinks of mixed motives, a natural direction of study (or speculation, as the case may be...) is that of determining all possible extensions (or iterated extensions) between two motives. And that's what motivic cohomology is, essentially: the study of these Ext groups. More formally, every variety $X$ should determine an object $C(X)$ in the bounded derived category of mixed motives, collecting together all the various mixed motives $H^i(X)$, and the $(i,j)^{th}$ motivic cohomology of $X$ is (up to twisting conventions) the abelian group of maps from the unit object to $C(X)$ \ $[i](j)$ (the $j^{th}$ Tate twist of the $i^{th}$ shift of $C(X)$) in the derived category of mixed motives.
Now, there are a few points to make here. The first is that, though the above motivation and definition of motivic cohomology rely on an as-yet-conjectural abelian category of mixed motives, one can, independently of any conjectures, define a triangulated category which, as far as anyone can tell, behaves as if it were the bounded derived category of this conjectural abelian category. The most popular such definition, because of its simplicity and relative workability, is Voevodsky's. So the basic theory and many basic results on motivic cohomology are unconditional.
Another thing to say is that, as always, matters with motives are illuminated by considering realization functors. Let me single out the $\ell$-adic etale realization, since its extension from pure to mixed motives is straightforward (unlike for Hodge structures): any mixed motive, just as any pure motive, yields a finite-dimensional $\ell$-adic vector space with a continuous action of the absolute Galois group of our base field. It then "follows" (in our conjectural framework... or actually follows, without quotation marks, in Voevodsky's framework) that the $(i,j)^{th}$ motivic cohomology of X maps to the abelian group of maps from the unit object to $C^{et}(X)$ \ $[i](j)$ in the bounded derived category of $\ell$-adic Galois representations. But this abelian group of maps is just the classical (continuous) $\ell$-adic etale cohomology $H^i(X(j))$ of the variety $X$, making this latter group the natural target of an $\ell$-adic etale "realization" map from motivic cohomology.
So here comes the third point: note that this is the etale cohomology of $X$ itself, not of the base change from $X$ to its algebraic closure. So this etale cohomology group mixes up arithmetic information and geometric information, and the same is true of motivic cohomology in general. (Think especially of the case $X=pt$: the motivic cohomology of a point admits a generally nontrivial realization map to the $\ell$-adic Galois cohomology of the base field.) For example, it is expected (e.g. by Grothendieck -- see http://www.math.jussieu.fr/~leila/grothendieckcircle/motives.pdf for this and more) that for an abelian variety $A$ over an ``arithmetic'' base field $k$, the most interesting part of the motivic cohomology $H^{(2,1)}(A)$ (again my twists may be off...), by which I mean the direct summand which classifies extensions of $H^1(A)$ by $H^1(G_m)$, should identify with the rationalization of the abelian group of $k$-rational points of the dual abelian variety of $A$, the map being given by associating to such $k$-rational point the mixed motive given as $H^1$ of the total space of the corresponding $G_m$-torsor on $A$. And in this case, the above "realization" map to $\ell$-adic etale cohomology is closely related to the classical Kummer-style map used in the proof of the Mordell-Weil theorem.
So in a nutshell: motivic cohomology is very related to motives, since morally it classifies extensions of motives. But it is of a different nature, since it is an abelian group rather than an object of a more exotic abelian category; and it's also quite different from a human standpoint in that we know how to define it unconditionally. Finally, motivic cohomology realizes to Galois cohomology of a variety itself, rather than to the base change of such a variety to the algebraic closure.
Hope this was helpful in some way.