Wait is used to talk about letting time pass.
I've been waiting for you for hours.
I don't have time to read this book now - I'll have to wait for the holidays.
Expect is used to talk about things you believe are going to happen.
I've been expecting you (= I knew you were coming)
I expect I'll get a promotion next year.
She's expecting a baby.
Now over to you:
What are you expecting to happen this weekend?
Is there someone you know who you always have to wait for?