I installed on a pretty much brand new install of Windows, and ran into a startup problem (failed to start, error # blah blah blah) but this post in the game forums solved it for me. Posting here in case anyone else runs into this.

http://www.ftlgame.com/forum/viewtopic.php?f=9&t=2007&sid=f036cc807530b5239ef5df9b364cb733

For anyone who would like to know the specific error and its solution without jumping to the other site:

Error message: The application failed to initialize properly (0xc0150002).
Solution: Download the "Microsoft Visual C++ 2005 SP1 Redistributable Package"

Link to get this package:
http://www.microsoft.com/en-us/download/details.aspx?id=18471

It worked like a charm in my case.