aggravate
People who chew with their mouths open often aggravate the people near them, meaning that they exasperate their neighbors.
irritate
To irritate is to cause an unsettling reaction, whether it's of the body or the nerves. You can irritate someone by talking too loudly on your cellphone in public. You can also irritate your skin, causing a rash or redness.
I get aggravated at irritation but not necessarily in the reverse order. I begin to get aggravated after a while but irritation comes on immediately.