I think you have done some mistakes when writing down your equations. An implicit ordinary differential equation can be written:
$$f(t,x,x') = 0$$
and a DAE can be written:
$$f(t,x,x') = 0$$
and since these two equations are syntactically equal, it is very easy to be confused about the distinction.
First, I want to make one thing clear. When talking about ODEs it is quite alright to talk about just one equation, e.g. the equation
$$y'(t) + y(t) + t = 0$$
is an ordinary differential equation.
But when talking about a DAE, you always, in a non-trivial case, talk about a system of equations. If your DAE contains only one equation it will either be a differential equation or an algebraic equation (in this context, algebraic means not containing any derivatives). Thus, in the rest of the post, $f$ and $y$ will be vectors of functions.
Thus, the difference between an implicit ODE system and a DAE system is, in a way, that the DAE system can contain purely algebraic equations and variables. The more technical and correct criterion is that the Jacobian $$\frac{\partial f(t,x,x')}{\partial x'}$$ needs to be non-singular for the system $f(t,x,x') = 0$ to be classified as an implicit ODE.
To make the distinction more clear between a DAE and an implicit ODE, you can split the vector $x$ in two parts, $x_D$, containing the $x$ for which derivatives occur in the DAE and $x_A$, containing the algebraic $x$, i.e. the $x$ for which no derivative occur in the DAE, and we write the DAE on the form
$$f(t,x_A,x_D,x_D') = 0.$$
we can also split the function $f$ into two parts: $f_D$ containing the equations containing derivatives and $f_A$ not containing any derivatives.
A classic example of a DAE is the following formulation for the motion of a pendulum:
$$\begin{align}
0 &= x' - u \\
0 &= y' - v \\
0 &= u' - \lambda x \\
0 &= v' - \lambda y - g\\
0 &= x^2 + y^2 - L^2
\end{align}$$
where $L$ (the length of the pendlum) and $g$ (gravitational acceleration) are constants. Classifying the variables as differential (belonging to $x_D$) and algebraic (belonging to $x_A$), we see that $x,y,u,v$ are differential and $\lambda$ is algebraic. All equations except the last (the length constraint) are differential.
We can calculate the Jacobian of this system. We order the functions as above and the variables as follows: $(x,y,u,v,\lambda)$. Then the Jacobian will be:
$$\begin{pmatrix}
1 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0
\end{pmatrix}$$
which is singular since the last row ad the last column is zero, hence the system is a DAE.
One often wants to apply techniques to a DAE to transform it to a semi-explicit DAE of index 1, which can be written as follows:
$$\begin{align}
x_D' &= g_D(t,x_D,x_A) \\
0 &= g_A(t,x_D,x_A)
\end{align}$$
because then $g_A$ can, in theory, be solved for $x_A$, which can then be inserted into $g_D$, which can then be integrated numerically.
The pendulum example might look as it is on this form, but its index is not 1.
These are two distinct and not very related concepts (though not completely unrelated).
Differential algebra is the study of differential rings and fields and related structures. Let me briefly mention some things about differential fields to give you some idea about what differential algebra is about.
A differential field is a field $\mathbb F$ together with a function $\partial : \mathbb F \to \mathbb F$, called a derivation, that satisfies the product rule:
$$\partial(xy) = \partial(x)y + x\partial(y)$$
An element $x \in \mathbb F$ is called a constant of the differential field if
$$\partial(x) = 0.$$
The set of all constants form a subfield of the differential field.
An example of a differential ring is $\mathbb R(t)$, the field of rational functions in $t$ over $\mathbb R$, with the derivation $\frac{d}{dt}$, differentiation with respect to $t$. The constants of $\mathbb R(t)$ is $\mathbb R$.
Elements of differential algebra are used in e.g. differential Galois theory and symbolic integration.
A differential algebraic system of equations is a system of equations where some equations are algebraic equations and some are differential equations. The equations need not be polynomial. I say system of equations, because if it is not a system of equations, i.e. there is only one equation, it will either be purely algebraic or differential.
An example of a DAE system is the equations describing the motions of a planar pendulum, having position $(x,y)$, velocity $(u,v)$, all functions of time $t$, with length $L$:
$$\begin{align}
\dot x &= u \\
\dot y &= v \\
\dot u &= \lambda x \\
\dot v &= \lambda y - g \\
L^2 &= x^2 + y^2
\end{align}$$
as you can see, the last equation is algebraic and not differential.
I have explained the difference between an ODE and a DAE here: What is the difference between an implicit ordinary differential equation and a differential algebraic equation?
As an aside, your description of DAEs ("polynomials with complex coefficients and the unknown variables are $z,x,x'$") got me thinking of holonomic functions, and while they are not exactly what you described, they do come close. A holonomic function $y(t)$ is a function that satisfies
$$a_r(t) y^{(r)}(t) + a_{r-1}(t) y^{(r-1)}(t) + \dots a_1(t)y'(t) + a_0(t)y(t) = 0$$
where each $a_i(t)$ is a polynomial in $t$.
Best Answer
DAE vs. ODE
Almost any DAE system can be reduced to an ODE system. As this requires derivatives of the equation, the equations themselves have to be differentiable to the required order.
$\newcommand{\pd}[2]{\frac{\partial#1}{\partial#2}}$ In your example, you could, as per comment, solve the second equation for $y$ and insert into the first one. This is the same as taking the derivative of the second equation to get a differential equation for $y$, $$ \pd{g}{t}(x,y,t)+\pd{g}{x}(x,y,t)\cdot f(x,y,t)+\pd{g}{y}(x,y,t)\cdot \dot y=0. $$ As is visible, and also demanded by the implicit function theorem, this only works if $\pd{g}{y}$ is invertible. If that is not the case, further derivatives of the equations may give rise to a complete ODE system, the maximal number of necessary differentiations of any equation is the index of the DAE.
Consequently, any ODE system is an index-0 DAE system.
This process towards an ODE may fail, either because the equations are not smooth enough like in $x_1'=x_2,~ x_1=q$, when $q$ is not differentiable. But also the process of index determination can fail to stop, that is, there is no differentiation order at which one can extract explicit equations for the highest order derivatives. In other words, there may not be any consistent system state, consistent with all equations and their derivatives.
Usefulness of DAE
Especially physical systems can be encoded more closely to the physical description, the first principles, using DAE systems. This enables software like modelica where large systems are constructed from basic building blocks having an inner dynamic of their state and pins/variables connecting to the outside and other building blocks.
For instance, consider the pendulum as mechanical system restrained to a circle, \begin{align} \ddot x+\lambda x&=0 \\ \ddot y + g/m + \lambda y &= 0 \\ x^2+y^2-r^2&=0 \end{align} or the corresponding first order system. While the algebraic equation is solvable for one of the variables, this will not give a dynamical equation for the Lagranian variable $\lambda$, one needs 2 derivatives to eliminate $\lambda$ and $3$ derivatives of the equations for an ODE for $\lambda$.
This system directly expresses the physical situation in Cartesian coordinates, containing the gravity force as gradient of the potential and the gradient of the surface with its multiplier as virtual force. While mathematically simpler, the transformation to polar coordinates as in the reduced pendulum equation loses this direct physical context.