Your following two sentences have bullet points, followed by an explanation of what they mean.
- I fought with my brother for the bed.
Here there are at least two possible meanings: 1) You and your brother joined together to fight an unnamed third person for possession of the bed; or 2) You and your brother argued back and forth--perhaps even to the point of exchanging blows--for possession of the bed (or at least for the privilege of sleeping in it for a night or more, for example).
Years ago, there was a quaint expression in American English. I'll adapt it to the context of fighting for a bed: "I'll wrassle you for the bed." Translation: I'll wrestle with you, and the winner of the wrestling match gets to keep the bed. (The loser would signify having lost the match by saying, "I give up," or some such admission of defeat.)
- I fought with my brother over the bed.
Here again there are at least two possibilities: 1) You and your brother argued back and forth about the bed, but not necessarily for the ownership of it, but for some other unspecified reason (e.g., over whether it was an 18th-century antique or a 20th-century knockoff of an antique bed); or 2) You and your brother banded together to fight some other relatives for the ownership of your late father's antique bed, for example.
In each case, the bed is the object over which you are fighting, whether in sense number 1 or sense number 2. A further wrinkle, however, which was suggested by a commenter below my answer, is that by saying "I fought my brother for the bed," any potential third party is hereby excluded from the fight. It's just mano a mano! (i.e., hand-to-hand combat between brothers).
As for whether the use of the word for or the word with makes a difference in meaning, well, you be the judge.
By the way, in light of your use of the sentence "Or is it all semantics?", I suggest that every discussion about word choice involves semantics. More often than not, the discussion centers around either 1) what a word means to you and how that meaning might be different from the meaning I attach to it; or 2) what the connotation of the word is and what it means to each of us. Either way, it's a matter of semantics, yes?
For that reason, whenever I hear a comment such as, "Meh, it's just a matter of semantics," I say, "Well, duh, isn't everything a matter of semantics?"