Are differential-algebraic equations more expressive than ordinary differential equations? I am interested in systems of differential-algebraic equations (DAE), i.e., systems of equations of the following form
$$\dot{x} = f(x,y,t)\\0 = g(x,y,t)$$
I am confused about their relation to ordinary differential equations (ODE):
Are there functions that can be described by DAEs but not by ODEs? I.e., functions that are a solution for some DAE but not for any ODE?
 A: 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.
