The PGF manual states (p20 in version 2.10):
The pstricks package is certainly powerful enough to create any conceivable kind of graphic, but it is not portable at all. Most importantly, it does not work with pdftex nor with any other driver that produces anything but PostScript code.
Compared to pgf, pstricks has a broader support base. There are many nice extra packages for special purpose situations that have been contributed by users over the last decade.
The TikZ syntax is more consistent than the pstricks syntax as TikZ was developed “in a more centralized manner” and also “with the shortcomings on pstricks in mind.”
Note that a number of neat tricks that pstricks can do are impossible in pgf. In particular, pstricks has access to the powerful PostScript programming language, which allows trickery such as inline function plotting.
On the previous page, when discussing frontends to PGF, it says:
It would be possible to implement a pgftricks frontend that maps pstricks commands to pgf commands. However, I have not done this and even if fully implemented, many things that work in pstricks will not work, namely whenever some pstricks command relies too heavily on PostScript trickery. Nevertheless, such a package might be useful in some situations.
I remember reading something that Till Tantau wrote on a mailing list about this; sure enough, a search for "pgf pstricks" brings up this message from 2008 (I'm not sure if it is legal to cut-and-paste the whole message here, all of it is relevant I deem) which starts with:
actually, that passage in the manual is rather old, TikZ can now do
most of the things that pstricks can.
He goes on to say that the major advantage of PSTricks is that postscript is better at maths than TeX and that PSTricks can do fancy things with text that isn't possible in PDF. The last comment is:
According to user feedback, it seems to be an advantage that pstricks
has a short manual. A good book on TikZ would certainly help, here.
Or a good website ... such as this one!