"Was Sie betrifft, bin ich zuständig."
Yes, I would say there is a difference in German, but both "zuständig" and "verantwortlich" can be translated as "responsible".
zuständig reminds me more of a situation in some kind of office, etc. meaning something like "it's my job to do that" / "it's my job to take care of your problems".
responsible seems a little stronger to me, like "I take responsibility for that. If it goes wrong, you have to blame me" / "I am responsible for you, if you do something wrong, they will blame me as well".
Maybe that helps a bit.
Another translation that makes very little sense in English. such a sentence and with its aggressive sentiment simply would not be used. The nearest I got and I lost a heart, was 'What concerns you, am I responsible? However the translation seem more of an instruction or statement of a fact, about the work the person does and who is responsible for that work, rather than an inquiry about someones welfare.