Apposite and Opposite have quite different meanings.
Both words refer to position—though they also have metaphorical senses that go far beyond position.
While you may know that opposite means “on the farthest away side” or “facing” position-wise, you might not know that apposite means “next to” or “touching” (in it’s position meaning)—although, to complicate things, it is almost never used this way:
Common: Conveniently, the court house is opposite (facing) the jail.
Rare: There is a playground apposite (next to) the school.
However, apposite is used a little more commonly in two ways. Metaphorically, apposite can be used to mean “appropriate” or “fitting.” Here is a more common example of apposite used in a sentence:
Your apposite comments about his character are quite welcome.
And another form of apposite—appositive—refers to a common English grammatical construction—a phrase that gives an alternative and non-essential description of something:
The house you saw earlier, the white one, belongs to my friend.
Pink Floyd released Dark Side of the Moon, a revolutionary album, in 1974.
The phrases in bold text above are called appositives (or appositive phrases).
Opposite also has a metaphorical meaning—“most different”—which you probably know already:
Is love the opposite of hate or the opposite of fear? I don’t know.
English speakers almost never use the word apposite in everyday conversation. Of course, the word opposite is quite common.