UPDATE: Just as suddenly and mysteriously as this functionality appeared, it's gone again.  If you didn't download games like Crash or Spyro before Sony disabled them, then you're out of luck as once again a large chunk of the PS1 and PSP digital library is unavailable to download and play on the Vita.  Any games you did acquire during this period still work, so don't delete them.  I only managed to get the first three Crash games, Spyro 2, and Castlevania Chronicles before the "problem" was "fixed".  Sony giveth and Sony taketh away...

While Sony's PlayStation Vita is able to play classic PlayStation and PlayStation Portable games, some games such as the popular Crash Bandicoot and Spyro the Dragon are unable to run on the Vita for a variety of technical and/or licensing issues.  That's changed now as Sony recently (and quietly) made nearly all available PS1 and PSP games playable on the Vita.  Check your download lists!  PlayStation Lifestyle summarizes things:

While there isn’t a complete list of titles that now work with the PS Vita, we do know the Crash and Spyro PS1 titles in the recent Flash Sale are compatible, with reports also showing Jeanne D’Arc, Metal Gear Solid: Portable Ops, Brave Story, Motorstorm: Arctic Edge, and Phantasy Star Portable 2 now work as well.

While I've had the Crash and Spyro trilogies on my PS3 for a while now, I haven't done much with them.  I'm experiencing the sequels to those properties for the first time on the Vita now and am blown away by how much better they play on the small screen.  I've also gone back and dug some of my dusty digital PSP games from virtual storage to finally play games that were too large for my old PSP memory card such as LittleBigPlanet and ModNation Racers.  It took long enough for this update to happen, but it's finally great to have so much PlayStation history available on the go.  If only Mega Man: Powered Up was available digitally, I could retire my PSP outright.