There's no difference in meaning. "At a loss for words" is an idiom that goes back centuries. "Lost for words" is a more modern and informal equivalent.

They mean the same thing. In the US, I think the first way is more common. "To be speechless" is a third way. Unfortunately, in English as in other languages, there can be more than one way to say the same thing. It's often hard to explain why one is used rather than the other. For example, nobody has been able to explain to me how a Spanish speaker decides whether to say "hubiera" or "hubiese."

They mean almost the same but they feel different. To understand the difference, ignore "for words" and just consider the difference between "I am at a loss" and "I am lost". They do not have the same meaning. "I am at a loss" means I am stymied, foiled, blank, bare, emptied, stranded, forlorn. "I am lost" just means I cannot find my way. "At a loss" is a more powerful image.