Isn't picking a string the same whether is you pick down or up on a string?
yeah but you can do it twice as fast if you do it both ways.
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You can play much, much faster with alternate picking
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its less effort and will help later on when you need to play faster
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well firstly you can pick twice as fast, if you pick up AND down...

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1st is the speed that alternate picking offers
when you master it, youll use it even when slow and youll notice that playing solos or a harmony, whatever, just became easier =)
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You're going to get a very slight difference in sound between upstrokes and downstrokes. It's a very minute difference, but it's there. And as for alternate picking. if you want to be able to sound like a crazy shredder, gotta nail that alternate picking.
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Not only in terms of playing faster, alternate picking can help you by allowing you to save energy. This is commonly called efficiency picking. By learning certain picking patterns for scales you save energy over time. For instance, if you need to play three notes on the fifth string and then three notes on the fourth string, follow a pattern of DOWN, UP, DOWN. This way, when you hit the last note on the fifth string its on a downstroke, allowing you to quickly and efficiently move to a down, up, down pattern on the next string.
Ok, It's really a combination of what every1 has already said.

1. Alternate Picking gives you your accent, your downstoke is going to be stronger then your upstoke, your accent should be on your 1st and/or 3rd. alternate picking makes it easy on simple timing songs.

2. Speed wize it leaves downstrokes for dead.

3. its easier to move across strings once you get the hang of it.
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Alternate picking the the most important technique for a lead guitarist and the other techniques aren't even close (maybe bending and vibrato, but for speed, it's alt picking).

Yes, practicing it is hard and often boring, but it is necessary.

However, if you somehow managed to downpick 18 nps (but you can't), you would not need alternate picking.