The difference between the two words is very subtle, and in most cases they are interchangeable.
"Expect" is more common, first of all, and therefore has more casual meanings (for instance, to say a woman is "expecting" means that she is pregnant). For your question, expect is more concrete. Someone is expecting something, someone, a package, a certain outcome, etc. and they are somewhat confident that it will happen. If I expect you to join me for lunch, that means that we already made plans to meet up for lunch, and I will be surprised if you don't show up.
"Anticipate" isn't used as commonly. As mentioned by Rajesh, it can mean to act because you suspect something will happen. But it can also be a synonym for "expect." If you anticipate that something will happen, it means that you SUSPECT it MIGHT happen. Anticipation involves a lot more speculation. When I hear the word anticipate, I imagine a general or a chess player trying to figure out which way his opponent is going to move.
Also, as a general critique on the way your question is worded:
You put: Hi dear,
May I have your advise for below question , what the different between 'anticipate' and 'expect' ?
and by the way, is there any one could help me check whether the sentences is correct or not?
Thank you so much.
I would put this instead. The corrected words are in all-caps (meaning all capital letters), but this is just to show you where I've made corrections:
HELLO,
May I have your ADVICE on the QUESTION BELOW?
What IS the DIFFERENCE between 'anticipate' and 'expect'?
And by the way, is there ANYONE WHO could help me check whether the SENTENCE is correct or not?
Thank you so much.