I have a Droid Razr that has gone somewhat kaput on me today at work. It is unfortunately my work cell phone and is somewhat required for me to function in my line of work. Before posting, I'd spent much of today (as opposed to work I probably should have been doing instead!) searching and trying various and (hopefully not) sundry things in an effort to revive my potentially dead phone.
A few points to note:
I'm supposing at this point that there is something completely wrong with the phone, likely no hope of data recovery, and a week or more wait supposing that my $6/month fee for an insurance will not be rejected on account of <a reason>.
If that is the case, why then would it recognize when it's plugged into a computer USB port and nowhere else? It seems oddly specific..
It's a puzzler that the phone will respond only to being plugged into a PC USB. Has the phone suffered any kind of damage? Water? Hard drop? Battery completely drained to 0? Sticking buttons?
If you want to try again to recover your data and don't want to try for a replacement just yet, I'd try another charger and cable that I know are good (from a friends phone, etc.) Leave it plugged into a wall socket for 2-4 hours (I know you have already tried this, but try it again with a different charger and cable). The soft reset and factory reset via external buttons are a little tricky. It took me a few tries to work it out the first time that I tried it.
In the end, if trying one more time doesn't work, then bite the bullet and go for a replacement.