Are there any specialized Computer Algebra Systems (or packages for these) for finding closed form solutions to
a) partial differential equations, b) stochastic differential equations?
If yes, what experiences do you have with these?
I have used Wolfram Mathematica extensively in my undergraduate course so far. Although the PDEs and systems of PDEs I have encountered have not been overly complicated, Mathematica is able to solve them in closed form most of the time. While not a "specialised" CAS for PDEs/SDEs, it gave me the closed form solutions I was looking for.
This link may be useful in terms of gauging what you can do.
Most ODEs, nevermind PDEs and SDEs don't have what one would usually call "closed form" solutions.
Are you interested in a special class of equations (e.g. linear, constant coefficient, on flat manifolds without boundary) for which closed form solutions are likely to exist?
For usual (non-stochastic) PDEs indeed consider giving Maple a try. You don't even have to bother too much with the PDEtools package, just try Maple's command pdsolve and see for yourself.