They mean the same thing literally and you can interchange them, although to me there's one subtle difference.
If you say something is obvious, it can sometimes sound like you're implying that anyone who thinks otherwise or that hasn't realised it yet is stupid.
'Apparent' is much more neutral.
Their adverb forms are very different.
'Obviously' means exactly what you'd expect.
Use 'apparently' when you're reporting a fact that you can't personally confirm.
"One plus one is obviously two."
"Vegetables are apparently very good for you." (ie. You've heard that vegetables are good for you, but you haven't personally tested it)