To upvote we need a reputation of 15, but being able to downvote requires a reputation of 125. Why do we need more reputation to downvote?
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Voting a question or answer up is an act of pure goodwill. The only reason to do it is because you think it's a good and useful question. If you vote up junk that doesn't deserve it, so be it. A newcomer may not really know what deserves voting up, but they will probably not give upvotes to spam, or answers that say "I have this problem someone help me".
Downvotes are given for two reasons. The good reason is that this is a poor question or answer, and people shouldn't be fooled by it. Making the internet better. The bad reason is "I want to make my answer [or my friend's answer] look better and show up first". By asking you to earn some rep, the site is asking for some commitment to making the internet better. The hope is to weed out those who would downvote for the wrong reasons, or convert them to a sharing way of thinking.
125 rep is probably not quite enough to achieve that, but it's something.
To discourage things like downvoting answers that compete with you or your friends, a downvote comes at a cost of -2 rep. The cost has to be meaningful.
If you could downvote at 15 reputation, you could create a "troll" user. The troll would gain 3 upvotes and could then downvote 8 times.
The requirement of 125 makes it harder to create troll accounts. Few trolls are willing to invest this effort. And those few that make it through can be investigated by moderators. It's way easier to show a pattern of abuse with 13 votes as proof.
Given that downvoting detracts reputation from the recipient, it is not something you want to give to a one-time question asker that happened to get a few upvotes but hasn't really gotten the hang of how SO works. Also, at such a low level of reputation, you could quickly cause yourself to lose enough reputation to lose privileges, causing no end to confusion among the newly initiated.
There is a deeper answer to this, I'm sure, but I think these facts played a part in the decision.