Entscheidungsproblem vs. Unvollständigkeitssatz (soft question) The first term is used by Hilbert in his 1928 work, but in Gödel's later work, the same thing is referred to as Unvollständigkeitssatz ("incompleteness theorem"). For today's German CS researchers, it seems Unvollständigkeitssatz is more commonly used, and Entscheidungsproblem ("decision problem") is still understood, but not necessarily associated with das Halteproblem (which seems to be more common after Turing's work on automata). On the other hand, for English CS researchers, Entscheidungsproblem is usually the only word they are familiar with.

Note: the words are not the same, and it could be argued that Hilbert's question about deciding was answered in the negative for a particular case by Gödel's statements about incompleteness, so that incompleteness demolishes decision in general. 

Interestingly, when looking at the German Wikipedia, there is no entry for Entscheidungsproblem, but there is one for Gödelscher Unvollständigkeitssatz, and the entry about Hilbert uses Gödelscher Unvollständigkeitssatz. When looking at the English Wikipedia, one readily finds an entry for Entscheidungsproblem.

How come Entscheidungsproblem is no longer used in German?
The two words do not refer to the same thing. Hilbert's Entscheidungsproblem was the question whether there is an algorithm that decides the universal truth of first-order logical sentences, which was answered negatively by Turing in his famous 1936 paper "On Computable Numbers, with an application to the Entscheidungsproblem". The word literally means decision problem. I assume that the word is no longer used since it refers to a problem that has been solved. In English it may still be more common due to its prominent use in the title of Turing's paper.

Gödels Unvollständigkeitssatz is his incompleteness theorem, stating that no consistent arithmetic theory is complete, in particular it can not prove its own consistency. This answered negatively a different question of Hilbert, viz. the second of his 23 famous problems, which was to prove the consistency of the axioms of arithmetic.