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
Let $K$ be one of the following fields: the complex numbers, a finite field, a number field (and we could amalgamate the last two into the more general case of a field finitely generated overs its prime subfield).
In each case we can consider the category of smooth projective varieties over $K$, with morphisms being correspondences [added: modulo the relation of cohomological equivalence;
see David Speyer's comment below for an elaboration on this point]. (I am really thinking of the category of pure motives, but there is no particular need to invoke that word.)
In each case we also have a natural abelian (in fact Tannakian) category in play: in the complex case, the category of pure Hodge structures, and in the other cases, the category of $\ell$-adic representations of $G_K$ (the absolute Galois group of $K$) (for some prime $\ell$, prime to the characteristic of $K$ in the case when $K$ is a finite field).
Now taking cohomology gives a functor from the category of smooth projective varieties to this latter category (via Hodge theory in the complex case, and the theory of etale cohomology in the other cases). The Hodge conjecture (in the complex case) and the Tate conjecture (in the other cases) then says that this functor is fully faithful.
The consequence (if the conjecture is true) is that we can construct correspondences between varieties simply by making computations in some much more linear category.
Let me give an example of how this can be applied in number theory:
Fix two primes $p$ and $q$, and let $\mathcal O_D$ be a maximal order in the quaternion algebra $D$ over $\mathbb Q$ ramified at $p$ and $q$, and split everywhere else (including at infinity).
Let $\mathcal O_D^1$ denote the multiplicative group of norm one elements in $\mathcal O_D$.
Since $D \otimes_{\mathbb Q} \mathbb R \cong M_2(\mathbb R)$, we may regard $\mathcal O_D^1$ as a discrete subgroup of $SL_2(\mathbb R)$, and form the quotient $X := \mathcal O_D^1\backslash \mathcal H$ (where $\mathcal H$ is the upper half-plane).
We may also consider the usual congruence subgroup $\Gamma_0(pq)$ consisting of matrices in $SL_2(\mathbb Z)$ that are upper triangular modulo $pq$, and form $X_0(pq)$, the compacitifcation of $\Gamma_0(pq)\backslash \mathcal H$.
Now the theory of modular and automorphic forms and their associated Galois representations show that $X$ and $X_0(pq)$ are both naturally curves over $\mathbb Q$, and that there
is an embedding of Galois representations $H^1(X) \to H^1(X_0(pq))$. Thus the Tate conjecture predicts that there is a correspondence between $X$ and $X_0(pq)$ inducing
this embedding. Passing to Hodge structure, we would then find that the periods of holomorphic one-forms on $X$ should be among the periods of holomorphic one-forms on $X_0(pq)$.
Now the theory of $L$-functions shows that the periods of holomorphic one-forms on $X_0(pq)$ in certain cases compute special values of $L$-functions attached to modular forms on $\Gamma_0(pq)$. So putting this altogether, we would find that (given the Tate conjecture) we could compute special values of $L$-functions for certain modular forms by taking period integrals on the curve $X$. In certain respects $X$ is better behaved than $X_0(pq)$, and so this is an important technique in investigating the arithmetic of $L$-functions.
Now it happens that in this case the Tate conjectures is a theorem (of Faltings), and so
the above argument is actually correct and complete.
But there are infinitely many other analogous situations in the theory of Shimura varieties where the Tate conjecture is not yet known, but where one would like to know it.
There are also similar arguments where one uses the Hodge conjecture to move from information about periods to information about Galois representations and arithmetic.
I should also say that there are lots of partial results along the above lines in these cases; there are many inventive techniques that people have introduced for getting around the Hodge and Tate conjectures. (Deligne's use of absolute Hodge cycles, applications of theta corresondences by Shimura, Harris, Kudla, and others, ... .) But the Hodge and Tate
conjectures stand as fundamental guiding principles telling us what should be true, and which we would dearly like to see proved.
Summary: algebraic cycles are very rich objects, which straddle two worlds, the world of period integrals and the world of Galois representations. Thus, if the Hodge and Tate conjectures are true, we know that there are profound connections between those two worlds: we can pass information from one to the other through the medium of algebraic cycles. If we had these conjectures available, it would be an incredible enrichment of our understanding of these worlds; as it is, people expend a lot of effort to find ways to pass between the two worlds in the way the Hodge and Tate conjectures predict should be possible.
Another example, added following the request of the OP to provide an example just involving complex geometry: If $X$ is a K3 surface, then from
the Hodge structure on the primitive part of $H^2(X,\mathbb C)$, one can construct an
abelian variety, the so-called Kuga--Satake abelian variety associated to $X$. The
construction is made in terms of Hodge structures. One expects that in fact there should
be a link between $X$ and its associated Kuga--Satake abelian variety provided by some correspondence, but this is not known in general. It would be implied by the Hodge conjecture. (This constructions is discussed in this answer and in the accompanying comments.)
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