I can weigh in here, as I've owned/used a 62 for quite a few years and purchased a 63 a few months ago.
Let me first state overall, that this is my opinion regarding everything I'm about to say, so that I don't have to follow every line with a qualifier... Also, these basses both have TI flats on them, and always have. I have no idea of the tones/playability of these basses with rounds on them.
First off, these 2 basses aren't really comparable. Not any more comparable than say a 62 and an AM Standard P-bass. Seriously, different neck, different pickups, different finish... What else is there to make it a P-bass "different." The only things I can find that they both have in common (that aren/t common to all AM P-Basses) are:
1. They are both in the AVRI Line
2. They both have Reverse Tuners
That's pretty much where the similarities stop.
But if we must...
The 62.
As others have said, the neck is quite thin (front to back) and is really wide at the nut (1.75"!!). which at first takes some getting use to. IMO, the sound of the 62 is the most unique of all the P-basses, it's broad, airy, and has a dry/crisp(?) quality to it. You immediately know that you are hearing something different. I can totally see why Jamerson preferred to roll-on the bass on his B-15. Overall, I found I couldn't quite coax the tone out of it that I was wanting on the band-stand... That being said, in the studio, plugged in direct, the tone is heaven. Mine is so light that it has neck dive, which is a deal-breaker for me and ultimately relegated it to the studio.
The 63
Is the best Fender bass I've ever heard/played (and I'm not alone in this opinion). It's light, yet perfectly balanced. The tone is the P-bass tone I have in my head, yet sweeter somehow, on both ends. It's buttery and syrupy, but with a "weight" to the tone that is enveloping, and still never dark or cloudy. I use a lot of back-lines, which are sometimes pretty terrible amps (Line-6 the other day, for example) and so far I can't make the bass sound bad. The bass has the kind of tone and playability that make me a better player. We all know what that's like...
I'm not afraid to say it, the 63 is a better bass, overall, for me. Sure, the 62 has the mystique, and I'll even agree that it is the better looking of the 2, finish-wise, but honestly, can you tell a 62 from a 63 on the band-stand visually... (if that matters to you)... You might be able to tell by the amount of mistakes being made across that 1.75" neck!!

<grin>
Am I going to sell my 62? Of course not. Is it going to many see gigs, probably not. The 63 is my main gigging bass for the foreseeable future.
To the OP. If you couldn't tell from the post above. The 63 is what I would choose. No question.