What you've hit upon there is the fallacy of Moral Equivalence, or sometimes called False Moral Equivalence for clarity's sake.

As far as I know there's no Latin name, but that's what it is called when people attempt to equate two acts as morally similar when in fact it is unclear or unjustified as to why they are doing so.

If you are looking for the name of a fallacy which more generally attempts to call two consistent views contradictory, perhaps it could be construed as a re-phrased False Analogy fallacy. The underlying statement is that Capital Punishment is like Abortion in a relevant way, when in fact it arguably isn't.

The Latin name for it is tu quoque.

Like many logical fallacies, it is often misapplied - with this particular logical fallacy, it rather depends on context. In the case of moral statements, the fact that one does not adhere to it does not invalidate that one ought to. In the case of descriptive statements, the fact that the speaker does not adhere to it may be evidence that the descriptive claim is wrong.

Here is Evan Harper's example and a modified version:

1. S condemns homosexuality but has been known to frequent gay bars. Therefore, homosexuality is morally acceptable.
2. S believes sexual orientation/preference can be changed from homosexuality to heterosexuality using some method P, and S claims that they have successfully used P to change their own sexual orientation/preference. But S has been known to frequent gay bars subsequent to using method P. Therefore, we have reasons to doubt the efficacy of P, and perhaps to doubt the testimonial reliability of S (if S knows that he has been to gay bars, of course).

In case (1) we have a tu quoque, but I don't think we do in (2) even though both fit the argument schema on the above-linked nizkor.org page. I'm guessing most people are happy to grant that (1) commits the fallacy but (2) does not.

(As an aside, there may be valid defeaters for the conclusion in (2) - if visiting the gay bar was part of method P, or if it was explained away as a one-off lapse etc.)

It's an ad hominem, since it attacks the credibility of the arguer rather than the argument itself. Your specific example is problematic, but I see what you were getting at, so here's another example:

Mr. Smith condemns homosexuality, but has been known to frequent gay bars. Therefore, homosexuality is morally acceptable.

(In this case most skeptics would presumably agree with the conclusion, but the argument used to reach it is still an ad hom.)

I would say you were guilty of a double-standard, and I'm not sure it's a logical fallacy per se, just hypocrisy. You're applying one standard of logic to your opponent's argument but failing to apply it to your own.