I am not sure there is even a nuance. But if I want to push it and find a tiny bit of difference, I would say "egy bank" identifies it as "a bank", while if we just say "bank", then it ALMOST sounds like an adjective. So, it would refer to the functionality of the building, as a characteristic of it. What it is used for. What it serves as.

"A régi épület egy bank." - The old building is a bank.

"A régi épület bank." - The old building serves as a bank. It functions as a bank.

Does this make sense?

Let's assume that the building has served as a bank for a hundred years. But various banks came and went during the years. So it has always served as a bank in the last hundred years, but it changed owners/tenants several times. And of course it is a bank now, as well.
"Az elmúlt száz évben mindig bank volt, és most is (egy) bank."

Maybe the same tiny difference exists with professions, and probably many other sentences:

"Én tanár vagyok." - I serve/work/function as a teacher. My profession is teacher.

"Én egy tanár vagyok." - I am a member of the group of people who function as teachers. I am an instance, a specimen, of the definition of "teacher".

But this distinction may not actually exist at all, or not consciously. You may be better off just learning something like "we do not use 'egy' with professions". If you know a little Spanish, you may lean on that knowledge when deciding when to use "egy", as there are some similarities between the two languages in this regard. And also in the use of definite articles.