In modern English, these two words have totally different meanings. If you appreciate something you understand its importance and significance. For example, if you want to thank someone for helping you, you show that you are grateful by saying: 'I really appreciate all the help you've given me.' To estimate is to guess. For example, you don't know exactly how long it will take you to complete a piece of work, but, judging by how much work needs to be done and how quickly you usually work, you might estimate that it will take you about three weeks. In the past, these words were more closely related, but nowadays they are used in completely different ways.

"Appreciate" has several meanings. Here is an example of one of them: "I appreciate all your hard work" "Estimate" can be used slightly differently in different contexts. Here is an example: "I estimate that we will arrive in 20 minutes" You may think that the words are more closely related than they really are. For example, I think in Spanish "estimar" and "apreciar" can be synonymous. (Spanish speakers are free to correct me!) If you want a more useful explanation, I suggest that you give us some context and try to use them in your own sentences. You could then post them here for feedback.