Entscheidungsproblem vs. Unvollständigkeitssatz (soft question)
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.