In the context of your example sentence ("I will take advantage of/make the most of the opportunity to learn Spanish"), there is little difference.
However, "take advantage of" can carry connotations of exploitation (in the sense of abuse) and "make the most of" can carry connotations of doing the best you can in a bad situation. For example,
I will take advantage of my wife's broken leg by reading War and Peace to her
means "Bwahaha! She can't run away so I can do whatever I want!", whereas
I will make the most of my wife's broken leg by reading War and Peace to her
means "It sucks that she can't leave the house but that gives us the opportunity to spend some quality time together and share some great literature."
So, going back to learning Spanish, there's no abuse in learning Spanish on your month in Spain, so there's no problem saying "take advantage of". I suppose it's possible that people might imagine that you mean "It sucks I have to go to Spain but, hey, at least I'll get to learn Spanish." However, the very first thing you said is that you're looking forward to being there, so you obviously don't think it sucks, so nobody will think you mean that. It's obvious that what you really mean is "Spain is going to be great! And, as well as the fantastic weather and amazing food, I get to learn some Spanish, too! Awesome!" (Especially if you're talking to somebody and they can hear your positive tone of voice.)
If we consider an alternative situation where you've been asked to host a Spanish person for a month, you'd have to be a bit more careful. Taking advantage of their presence to learn Spanish would suggest you were going to bug them to teach you Spanish the whole time, whereas making the most of their visit by learning Spanish could suggest you were reluctant to host them. But, as with the example of going to Spain, nobody would think you were unhappy if you sounded happy.