Note: Because you took the liberty of asking your question in English, too, and because it might be found by people whose German is not too good, I’m going to provide an English answer.
The words Computer and Rechner are full synonyms of each other when used to describe the big machine that works with electricity. Unfortunately, that doesn’t work for all of their compounds: Taschenrechner, Großrechner but Computernutzer. The ones with the other word are possible but anything from ‘sound weird’ to ‘mean something different’. However, there are rare cases, where even the compound can be formed with both: rechnerunterstützt and computerunterstützt.
While the etymology of Rechner is not hard (something that does calculations for you), Germans usually wouldn’t know that Computer also derives from Latin computare and the English word was used throughout the ages for a profession known as Rechner in German. They didn’t mix because there was no need for anglicisms then.
While the two words are interchangeable, there can be reasons to use one or the other preferably. For one, many people dislike anglicisms and would prefer to use Rechner all the time. Similarly, when writing an essay in German class, Rechner might just fit in nicer with the surroundings. On the other hand in computer sciences (the area in which the OP said in comments his work is going to be handed in to), Computer might be preferred because it is almost like a technical term.
