Which is correct? "Confident in" or "confident of"?

To me, they sound interchangeable.

So, "I am confident of my abilities." "I am confident in my abilities."

However, I'd like to know from people here at English Stackexchange as well. Is there a reason to use one more than the other?

They are not interchangeable, strictly speaking. Loosely used, or in specific defining contexts, the senses do overlap.

Confident, adj. and n., from the OED

2. a. Having strong belief, firm trust, or sure expectation; feeling certain, fully assured, sure.
   ...
   c. Const. of (formerly with inf.).
   ...
   d. Const. in. (This has affinities with 3.)
   ...
3. Full of assurance, self-reliant, bold; sure of oneself, one's cause, etc.; having no fear of failure.

The meaning in isolation (without context) of 'confident of' conveys the sense of 'belief in', so your example

I am confident of my abilities

to me conveys that you believe in your abilities, but not necessarily that you act as if you believe in your abilities.

So, in contrast, your example

I am confident in my abilities

conveys more of a sense that when those abilities are in play, you act confidently.

The distinction I've drawn between the two echoes what the OED defines, perhaps more lucidly, by parenthetically mentioning the affinities of 2d., the construction with "in", with sense 3.

In some contexts I would use one, in others the alternative.

I feel sure he will pass his exam, I am confident of that. Why? Because I have full confidence in his teacher.

I would use confident of for expressing trust about facts or information, but confidence in for indicating my trust in a person, a group of persons, or a procedure. But note that in the latter case the idiomatic form is I have confidence in..., rather than I am confident of...

Where the noun "confidence" is used - it will almost certainly be "in" which follows. And in the two examples the OP gives, it would be more idiomatic to say:

I have confidence in my abilities.