All share the sense of 'be awaiting something to happen'.
'Hope', 'wish' and 'look forward to' almost always denote a desire for that, but 'expect' doesn't necessarily have to.
e.g. I expect him to beat me.
You don't want him to beat you, I supose, but you believe it'll happen, you're awaiting it to happen, you're expecting it.
Also, they have differences in formality and activity aspects.
'Expect' being the most passive of them, oftenly used when you just await.
'I expect he takes what he deserves. I will stay here, watch and laught at his face.
but: 'I expect to win this match, so I'll not give up.'
'Look forward to' is more active, as denote desire. But still oftenly used when you just await or do little to help out.
"I look forward to see you again, so, here, take my cellphone."
"I look forward to be with you at the end of the year."
'Hope', on the other hand, can be more commonly used with an active behavior as it is oftenly used for permanent or longer-term desires. You wouldn't just sit back at the sofa and don't try to fulfill your greatest desires, would you?
"Since I hope that I can get a promotion, I'm working harder and harder everyday."
but: "I hope the global warming can unite all men."
'I hope it rains today.'
And 'wish' can be the most compelling of them, as it's oftenly used for immediate or shorter-term desires. But, still, you may wish and do nothing about that.
'I wish to buy that game, dad. Give me money, please, please.'
'I wish to me the greatest warrior of the world. Get out of my sight, I have many battles ahead.'
but: 'I wish someone can do something about that broken sink. Man, it's disturbing. I can't sleep at night.'
Hope I could help you. =]