Ryan has explained it well. the meanings are as follows : assess (Verb) Evaluate or estimate the nature, ability, or quality of: "the committee must assess the relative importance of the issues". Calculate or estimate the price or value of: "the damage was assessed at $5 billion". estimate (Verb) Roughly calculate or judge the value, number, quantity, or extent of. (Noun) An approximate calculation or judgment of the value, number, quantity, or extent of something.

The meanings are very different and I am sure if you translate them in to Chinese or use a dictionary, you will be more clear, they translate completely differently. assess: 评估, estimate: 估计. Native English speakers use a dictionary all the time, you should get in the habit too. Imagine a house just had a big fire. First you go to assess the damage - was it a big or small fire, assess how much was damaged, what was damaged... After you make an assessment, you can estimate the likely cost of the damage. Estimate is like a slightly calculated or informed guess.

Correction: What is the difference between "estimate" and "assess"? Estimation is just a rough guess, not too precise at all. To assess the value of something is to come up with a much more exact number.