They are verbs that are often used when describing situations in which someone/something makes you feel angry. The nuance is slightly different.
"ムカつく" is a colloquial style of "腹が立つ". It is used when someone/something pisses you off.
”イライラする is similar to "うざい". It is used when describing situations in which someone/something is bothering you. It often indicates that you are panicking/frustrated because you can't do what you want to do.
Ken and Leo are talking about their colleague.
- Ken: Mike is smart and works hard. I don't know why, but I'm pissed off with the way he speaks.
マイクは頭が良いし、働き者だけど、なんかアイツの話し方ってムカつく。 - Leo: I know how you feel. His way of talking is kind of bossy.
わかる。偉そうなんだよ。 - Ken: Could be. And I don't know why he tries to talk with Kansai dialect even though he is from Tokyo. It's annoying.
そうかも。東京出身なのに、なぜか関西弁を喋ろうとするし。ムカつく。
Meg is complaining about her boss.
- Meg: I’m so irritated with Jeff. I have so many things to do, but he keeps asking me for favors.
あー、イライラする。やることが山ほどあるのに、ジェフは次から次へと作業を依頼してくる。 - Amy: He's still doing that? I'll help you after I finish with my tasks. Could you hang on a bit? I'll be done in about 30 minutes.
まだ、作業を振ってくるの？(自分の)作業が終わったら手伝うよ。ちょっと待って。30分くらいで終わるから。 - Meg: Thank you.
ありがとう。