Your two sentences have essentially the same meaning. The first sentence, with "but" as a coordinating conjunction, is more conversational than the second. The second sentence, with "however" as an adverb (also called a transition adverb or a discourse marker), is more formal than the first.
The sentences are perfectly correct, with "but" beginning the second independent clause and "however" beginning a second sentence which is in contrast to the first.
It used to be that "but" was not supposed to start a sentence. However, Bryan A Garner in The Oxford Dictionary of American Usage and Style* remarks that this is a useless rule, and notes that many first-rate writers – including H.L. Mencken, Lionel Trilling, and Edmund Wilson – begin many sentences with coordinating conjunctions.
There is, also, an old rule that "however" should not be the first word of a sentence. However, this is refuted in The American Heritage Dictionary of the English Language in a usage note under "however":
"Although some grammarians have insisted that however should not be used to begin a sentence, this rule has been ignored by a number of reputable writers."
While your two sentences are fine, they could be written alternatively, like this:
I learnt French easily. But I didn't like my teacher. (This sentence might appear in a dialog.)
I learnt French easily. I didn't, however, like my teacher.
*The Oxford Dictionary of American Usage and Style. Oxford University Press. 2002
**The American Heritage Dictionary of the English Language. Houghton Mifflin Company. 1996