Well a 2x will knock two stops off the maximum aperture - so your 70-200/2.8 turns into a 140-400/5.6. Although that sounds like a similar spec to the other lens, the reality is there will likely be a few issues: slower/ less accurate AF, poorer handling due to a bunch of factors including focus gearing and loss of balance, and ultimately it will be less sharp wide open - quite likely unusably so. So you end up stopping down an extra stop/ stop and a half to get sharpness, which then exacerbates the problems inherent with a long, slow, not well balanced lens - ie camera shake, so you end up increasing the ISO to compensate, etc etc.
Given that the 120-400 has OS (IS/VR), and should be perfectly usable wide open, you might well end up with at least three or four stop advantage over the 70-200+2x. That could mean shooting at faster shutter speeds and much less shake, or ISO 400 instead of 6400 - either way giving publishable results rather than blurry 'record' shots.
I have a 1.4x that works quite nicely with my Sigma 150 macro, okay with my Sigma 50-150 (sharp enough but unwieldy without tripod mount) but its barely usable on my Nikon 300/4 - kills the AF and is unsharp wide open. If you do go that route make sure you match manufacturers and try to get the version matched to the lens.
