Inspired by this SO question.
Texmaker is cross-platform, working on Windows, Linux and Mac OS X while TeXnicCenter works only on Windows.
Texmaker supports Unicode, for TeXnicCenter it's been announced for the new version.
Texmaker provides a built-in pdf viewer and can launch external viewers. TeXnicCenter supports any external pdf viewer. Both provide forward and inverse search with pdf and dvi.
Note, for Texmaker there's a fork TeXstudio (formerly named TexmakerX) with some improvements.
Just a minor addition to Stefan's answer, but something that might not be mentioned in other places:
I like to use MiKTeX with the
-auxiliary-files option, which puts all the files created during compilation (except for the final pdf/dvi/ps) in another directory. Texmaker doesn't work well with this, i.e. compilation works, but it doesn't find the .log file in a remote directory, so errors and warnings can't be displayed. There is a feature request for this; I have no idea how quickly such things get fixed.
TeXnicCenter worked just fine with the
Since I now need the Unicode support that TeXnicCenter doesn't offer, I've settled for having the auxiliary files cluttering up everything, and ocasionally getting rid of them via
latexmk -c foo.tex.
Personally, I prefer Winshell when I'm working on Windows. A main sticking point is the possibility to better manage my papers by organizing all files for each paper into a project, so I can quickly switch between the papers and see which file belongs to which project (I like to use a large number of .tex files -- one per section/subsection, if applicable). WinShell also has a BibTeX manager, if you need it.
One drawback is that it's not open-source (but is completely free), so if you have strong feelings about this, you should probably look elsewhere.
I'm also checking regularly the progress on TeXWorks, which is under development, but has a built-in PDF reader and synchronized scrolling of both the source and the rendered document. It lacks a lot of other features that I'd like to have --- notably, file organization, that I mentioned above.