Difference between OTF (Open Type) or TTF (True Type) font formats?
Solution 1:
OTF is more likely to be a “better” font, as it supports more advanced typesetting features (smallcaps, alternates, ligatures and so on actually inside the font rather than in fiddly separate expert set fonts). It can also contain either spline (TTF-style) or Bezier (PostScript Type 1-style) curves, so hopefully you're getting the shapes the font was originally designed in and not a potentially-poorer-quality conversion.
On the other hand, if you're downloading free fonts from shovelware sites, you're unlikely to get any of that. Indeed, you may simple be getting a TTF font renamed to OTF.
Solution 2:
OTF is a more recent format than TTF, so OTF has some features that TTF doesn't. (Which is a moot point if the font's creator didn't use them.)
One note from personal experience however: depending on what you're going to be doing with these fonts, I've found it's much easier to get tools that work with TTF as opposed to OTF. If you're just using them for desktop publishing / word processing, either will work fine, but if you're going to be doing anything programmatic, I'd recommend TTF just due to the higher number of tools / libraries out there.