String through body or bridge.
 

I recently got a American fender jazz. My first. I've always played mims and never had the chance to run the strings though the body. So when I set it up I did just because I could. But now I wonder what's the difference and why do people do it both ways?

Can't say I can tell much difference.

Personal experience with string-throughs (though not Fender) has been tone with more body (pun halfway intended) and more sustain.
To me, it's the same concept as an aftermarket bridge (on non-string-thru designs) such as Badass or Babicz or even Fender's HighMass; more material on each end of the strings=more sustain and ballz.

At the other end of the (string) scale, there's a couple of companies that use metal flourishes on the end of the headstock, and I think it's Fender that has a clip-on aftermarket headstock piece to add mass (and sustain), though I'd have to expend the enormously exhausting task of checking Google or Ebay for the name of the piece.

It's called the "Fat Finger".

Interesting...my sustain champ is my MIM FSR Jazz which is string-through-bridge. That sucker will hold any note until you tell it to stop. So I dunno. Can you imagine if it had string-through-body?? Perpetual motion! :D

My Ibanez ASB140 w/ Darkstar is pretty sustainy.