Irritating is the effect of an object that causes a certain degree of discomfort..for example a tiny stone in your running shoe..or the dust or an eyelid irritating your eye..
Annoying is a situation an act of a person that touches a nerve in you..like the little kids when they are impatient to reach a destination and ask "Are we there yet?" as in the little kid in The Mummy while talking to his captor inside the train..
My definitions only..others may see them differently
I liked DzyMsLizzy's answer-- I agree that they are synonyms, but they do have a slightly different meaning for me. Irritating is something that makes me mad, annoying is something that makes me irritated.
There's no "irritating orange" on youtube. Irritating is like an itchy rash. annoying is simply irksome, or bothersome. A case of poison ivy would not annoy me, it would however irritate me. Bad singers are annoying. fingernails scraping across a chalkboard are irritating.
I agree with you, fingernails scraping across a chalkboard are irritating. the noise of the nails being scrapped on the chalkboard, just imagine! Do you find it irritating scrapped fingernails on Paper? It's horrible :\
Irritating - that person beside you on the bus who won't move in to give your right leg some chair room; after all, their handbag needs a spot on the seat.
Annoying - getting all dressed up and ready to go out only to be texted 40 minutes before you're set to go and told, "really sorry, I don't feel well, mind if we skip it?" when you know the persons not sick.
For me, irritating is something which is a consistent or continuing annoyance, gradually niggling, little by little. Annoying is a thing or person who annoys by one particular action or for one particular reason - it just is annoying! Does that make sense?!!