It's a complex and multi-layered topic, as I alluded to in my previous comment, but I'll do my best to answer this. I'm by no means an expert on this, and I'm a white English dude so far from qualified really... But I have read a lot on the topic in addition to my own gut reactions to these things.

So... First let me clarify that by "wearing someone else's culture as a costume" I am really talking specifically about people in wealthy Western countries wearing the cultural clothing of (almost always) historically colonised peoples from elsewhere. With that being said the first point to call attention to is a kind of dry economic one - the outfits you see being worn as a Halloween costumer are broadly mass produced by companies with no affiliation to the cultures they are imitating. They make huge amounts of money selling these costumes to Westerners like me, while giving nothing back to the people they've taken them from. This follows a long and difficult history especially in the context of colonies - historically (not at all that this doesn't continue today) the West has plundered the world for all its worth, and this is just a relatively subtle modern example. So even before anyone puts the costume on I'm uneasy about it, personally.

The second point is specific to certain cultural garbs which are 'closed' within the cultures they come from. While the other reply points out that they, as a Mexican, don't mind seeing people dressed up in Mexican costume, you would be very hard pressed to find anyone Native American who is happy seeing anyone in a mass produced war bonnet. I won't pretend to understand the full significance of the headdress, but its well known that it is not something you just 'put on' if you are a Native American, and divorcing it from that cultural context both cheapens it and shows a general lack of respect towards the people whose clothing you're wearing.

I think that lack of respect is really the main part of my problem with costume-ising culturally significant clothing. Obviously there are clothes from all over the world which are just clothes, and quite likely the people who make those clothes would be delighted to see them being worn all over the world! But if you don't give enough of a shit to a) learn about the culture they come from and what the significance is and b) buy them from the actual people who created them, then you lose that connection and it ceases to be cultural exchange and becomes instead appropriation.