I wouldn't really consider it a competitor to Ableton Live, though there's some overlap. But, as a competitor to Pro Tools, it's hard for me to think of anything even in the same ball park as REAPER (and REAPER costs a fraction of what Pro Tools costs, as well). I use FL Studio or Renoise for the stuff that people use Ableton for; but, REAPER blows them all away for recording live musicians and then mixing the resulting tracks.
The term "DAW" has too many uses, and it can be confusing. REAPER is a Pro Tools competitor (and a good one). Ableton can be used for multitrack recording of live musicians, but I don't know of any studios that use it for that, unless their focus is on electronic production with some live music added on top. And, REAPER can be used for electronic productions and loop-based composition and live performance (and there's even a UI for REAPER that makes it act more like Ableton), but it's not its raison d'être.
If you're composing and performing predominantly electronic music in-the-box, then Ableton or others might be a better choice. But, if you're recording live musicians, REAPER is amazing. There can be some useful interoperation between them, as well. Live can plug into REAPER, so you can sync up your Live performance with your REAPER tracks, and produce a hybrid work. REAPER may be stronger for mastering, as well, depending on how you like to work.
Traditional recording engineers, who learned in a studio with tape and hardware mixers and patch bays, will be quickly comfortable with REAPER. The same can not be said of Ableton Live (or FL Studio, etc.). But, musicians who grew up on synths, sequencers and beat machines with sequencers (like 303, 808, etc.), may find the Ableton approach more intuitive. Both are high quality tools doing a very good job in their niche. REAPER just happens to be so good at such a low price that it stands out as exceptional and worth praise, IMHO.