Hello!
To annoy and irritate can both mean to make someone a little bit angry. For example, "It would annoy me if my brother were to ignore me"
and
"It would irritate me if my brother were to ignore me"
But both of these words have other meanings as well. "Annoy" can also mean to harm or attack repeatedly. For example, "the brave Saxon would annoy their coastline"
and
"Irritate" can also mean to cause discomfort or inflammation. For example, "Some chemicals can irritate dry and sensitive skin".
I hope that helps!