Do we use the word bedrücken more than betrüben for the verb to distress nowadays and has betrüben been abolished?
I don’t know where you got the information that betrüben had been abolished, but the following Ngram does not support that statement:
All three terms mentioned in Cryck’s answer are almost equally frequent in current usage.
The main difference between betrüben and bedrücken is their hidden meaning. Trüb means cloudy when referring to something that should be transparent (e.g. glass) or the weather. Thus, betrüben is the act of rendering somebody or someone’s feelings cloudy. On the other hand, drücken of course means to press so something that is bedrückend is something that is pressing down on you. Thus, a 16-ton weight would bedrücken you but hazy autumn weather would betrüben you.
While these nuances exist and are certainly relevant in literature, the two are pretty much synonyms in common usage.
Meaning of those two according to the Duden (a spelling dictionary of the German language):
Bedeutung 'bedrücken': auf jemandem lasten; traurig, niedergeschlagen machen
Bedeutung 'betrüben': traurig machen, traurig stimmen, bekümmern
Bedeutung 'bekümmern': betrüben; jemandem Kummer, Sorge bereiten
In the natural language you can use all of them equally as a translation of 'distress'.
'Bedrücken' and 'bekümmern' may suppress 'betrüben' in the long run but it is definitely not replaced/abolished yet. The most common one, based on what I have heard recently, would be 'bekümmern'.