Compare and contrast

When you are asked to compare two or more things, you generally reply by identifying their similarities and differences.  For example, you might compare two people by describing the ways that their looks, personalities, interests, etc. are similar AND different.  If you are asked to compare two cultures, you would describe how their foods, traditions, languages, religions, etc. are the same in some ways and  different in others.

As mentioned in the previous answer, it is less common to be asked only to "contrast" two items, but you might do that when you want to describe the ways that one item is better than another - if you are thinking of buying a new cell phone, for example.

Essay exams in English frequently ask you to "compare and contrast" two or more thoughts, ideas, approaches, etc.  If you are taking such an essay test, it is a good strategy to organize your essay by first identifying the similarities and then starting a new paragraph to describe the differences.

Remember that "contrast" is both a verb (to identify differences) and a noun ("there are many contrasts between our two cultures"...or "there is too much contrast in that photo", meaning extreme dark and light).  When used as a verb, the accent is on the second syllable (conTRAST).  When used as a noun, the accent is on the first syllable (CONtrast).

Hope some of that is helpful.

This is a really hard question for me to answer but I'll try to make the difference clearer.

'Compare' is used for similarities and differences. Usually, you can use 'compare' when you use 比べる。
'Contrast' is only used for differences. Usually, you would contrast two things.

Examples for compare:
1. I compared myself to my friend.
2. I compared my test marks with my friend's.
3. I compared the amount of candy I had with my friend.
4. We compared our lunches.
5. I compared these three cafes and I think this one is the best.
6. I compared Jack, Jill, and Jason and decided I liked Jack the best.
7. I compared myself to Benjamin and Jenny.
8. You're comparing apples to oranges. (Idiom meaning you cannot compare two things that are so different.)
You can also use 'compare' as an adjective meaning deserving of comparison.
1. I do not compare with her.
2. This book is beyond compare.

Examples of contrast 
1. I like the contrast of light and dark in this painting.
2. I like how the light contrasts with the dark in this painting.
3. This coffee, by contrast, tastes good. (Or, you could say 'this coffee tastes better.')
4. They have contrasting ways of thinking.
5. The happiness at the start of this song contrasts with the sadness at the end.

Sometimes, you can use both compare and contrast. Usually, people don't use the word 'contrast' very often and if you're not sure, it might be safer to use 'compare.' Feel free to ask me any questions.

Also, your question, 'How can I use separately these words?' is grammatically correct but it sounds a bit poetic and if you want to sound more like a native speaker, you should say 'how can I use these words separately?'
Usually, [adverb]+[verb] is used in the middle of the sentence and [verb]+[adverb] is used at the end.
E.g:
I ate the pie greedily.
I quietly read my book.