The Standard is made in Mexico and the American Standard is made in the USA... Basically that... According to some people on these forums the American is better constructed than the Mexican, and the American has an upgraded bridge, while I've heard that the Mexican has like a faulty bridge and noisy pickups, but honestly I've never owned neither of them... I only have a Squier...
I wouldn't go as far as to say that the (MIM) Standard has a 'faulty' bridge .. . it is just a simple angle bridge plate, with single groove saddles, not a thing wrong with it actually ... now, I have seen some installed slightly off that created a less than ideal scenario for string alignment, and as mentioned with the single groove saddle you are a bit limited, but the bridge is no different than what you have on your Squier (and a LOT of other basses), and it works just fine ...
I had some time to kill today on my journey into the big city, and hit both a GC and a Best Buy on an instrument browse ... the differences between the MIM Standard and American Standard arent all that impressive on paper, but I assure you, when played, examined, and handled all at the same time, the difference is VERY noticable ...
is it worth the extra money?? ... in my opinion, absolutely ... would I be willing to pay it?? ... perhaps ... but to play at church once a week, and to noodle at home, probably not ... but to really UNDERSTAND the difference, you need to spend some time with each ...
with all that said, when I got home today, after playing probably a dozen, Standards (MIM), some CV & VM Squiers, a Geddy, and Squier Jag, and the ONE LONE American Series Jazz on hand in BOTH stores combined, I was extremely happy with my stripped down, stock (except for flats) old Affinity Squier practice bass that I just leave out in the room for the cats to rub on when they want ... long story short, we become very used to what we have and play, and when set-up for our tastes, a ham sandwich can taste like prime rib against the rest of the menu ... I intended to compare my Classic Jazz to the AS Jazz I played today, didnt need to ... what I have sitting here, is here for a reason, they feel like home and are nice and comfy to me ...
long story short, just because it is better in every way (and the American Standard IS compared to a MIM Standard), it doesnt mean that you need to go that way ... I have less money into 3 basses than one American Standard would cost me, and I am just fine with what I have ... but, I would play an AS if someone asked me to on occasion ...

... JMHO