Expecting is rational. When you expect something to happen, you have a good reason to believe that it will happen. Hoping, on the other hand, is emotional. When you hope for something to happen, you would like it to happen, but you don’t know whether it will.

Compare:

She is expecting a baby. (= She is pregnant now.)
She is hoping that it will be a girl. (She doesn’t know whether it will be a girl.)

Notes
We can expect good or bad things to happen, but we only hope for good things.

Expect and wait
Wait does not mean the same as expect. You wait when somebody or something is late or when you are early for something.

Compare:
I expected her at eleven, but she didn’t come.
I waited for her till twelve, and then went home.

Grammar
Hope and wait take the preposition for before a direct object. Expect is used without the preposition.

I am expecting a phone call from Peter. (NOT I am expecting for a phone call from Peter.)
I am hoping for a raise. (NOT I am hoping a raise.)
I am waiting for John. (NOT I am waiting John.)

All three verbs can be followed by an infinitive.

I am expecting to hear from her.
I am hoping to hear from her.
I am waiting to hear from her.

Expect and hope can be followed by a that-clause.

I expect that he will arrive soon.
I hope that he will come up with some new ideas.

Wait cannot be followed by a that-clause.

Hope can be followed by a present tense with a future meaning.

I hope she doesn’t miss the train. (= I hope that she will not miss the train.)