What is the essential difference between ordinary differential equations and partial differential equations?

Please forgive my stupidity.

So many years after my undergraduate study and so many years after dealing with various concrete ODEs and PDEs, I still cannot tell the essential difference between them.

What specific belongs to PDEs but not to ODEs? What conclusion for ODEs cannot be generalized to PDEs?

At the moment, my understanding is simply that PDEs have more than one variables.

Both are differential equations (equations that involve derivatives). ODEs involve derivatives in only one variable, whereas PDEs involve derivatives in multiple variables. Therefore all ODEs can be viewed as PDEs.

PDEs are generally more difficult to understand the solutions to than ODEs. Basically every big theorem about ODEs does not apply to PDEs. It's more than just the basic reason that there are more variables. For an ODE, we can often view the single independent variable as a time variable, so that ODEs govern a motion or flow of an object in time. The idea of ODEs governing "motion" allows us to use many mathematical results that have analogues in physics (for example empirical behavior regarding Newton's law) and allow us to understand the solutions much more precisely.

Well, given a linear ODE, the set of solutions form a vector space with finite dimension. However, a linear PDE (like the heat equations) has a set of solution that form a vector space with infinitely many dimensions.

To see that, one may consider the ODE

$$ y'=-ay(t), $$ with solution, $$ y(t)=e^{-at}y_0, $$ (so, the vector space is one dimensional) Then the heat equation with periodic boundary conditions $$ u_t=u_{xx}, $$ has solution (use Fourier series/separation of variables) $$ u(t)=\sum_{k\in\mathbb{Z}}e^{ikx}\hat{u}(k). $$ We see that the linear combination has infinitely many terms, all them linearly independent, so, the vector space has infinitely many dimensions.

Linear PDEs may not have solutions. Hans Lewy constructed such an example about sixty years ago, and it probably surprised just about everyone in the field at the time. ODEs are much nicer in that regard. http://www.jstor.org/stable/1970121?&seq=1#page_scan_tab_contents

ODE has one Independent variable, say $x$. Solution is $y(x)$.

PDE has more than one independent variables say $(x_1,x_2,...,x_n)$: solution is $y(x_1,x_2,..x_n)$. Partial derivatives are in the equation. A partial derivative differentiates with respect to one independent variable (say $x_3$)while holding the other independent variables constant.