They can both mean 'on the contrary'. 'However' is more formal. I used to think that 'but' should not be used at the beginning of a sentence, but that rule is no longer really followed. "He hurt his leg, however it's not serious." "He hurt his leg, but it's not serious." (More often in spoken English) They both have separate meanings apart from the above definition: But = except "There was no one but Bob left to do the job." However = in any way "You'll have to get here however you can."

They both signify a counterpoint to a prior statement in a manner that provides further context by noting an exception, showing limitations, or decreasing the magnitude of what was said. Exception: "Dogs bark at everyone but their owner." "Dogs bark at everyone, however they will not bark at their owner." Limitation: "Ants are strong and can lift 50 times their weight, but they are still not strong enough to pick up a worm." "Ants are strong and can lift 50 times their weight. However, they are still not strong enough to pick up a worm." In terms of usage, I find that I will commonly use 'however' to start a sentence. 'But' is mostly used as a coordinating conjunction in the middle of a sentence. Both will require a comma after the word if they are starting the sentence. They will usually require a comma before the word if used as a coordinating conjunction combining two independent clauses. 'However' is more formal. In spoken language, 'however' will generally be given a longer break before it is said since it has become a more dramatic word than 'but'.

However can be used either at the beginning, middle or end of a sentence.

They basically have the same meaning, but 'but' is a conjunction, so you use it to connect two sentences (as I did in this sentence ;). It's like が・けど in Japanese. 'However' is more like だが・しかしながら, you often start a sentence with it.