Доктор or врач?

Врач is always доктор, but доктор might not be врач (for instance, доктор наук - Doctor of Science)

Доктор and врач are not completely interchangeable. When talking to a doctor you can call him доктор but never врач. Доктор also means a scientific degree (not only medical), while врач means doctor of medicine.

Those two words have different origin.

Врач is an old Russian word but доктор derives from Latin (that’s why it’s the same as in English and other Romanic languages)

And indeed, as noticed above, PhD is a доктор too.

When you are talking about a doctor of medicine it is usually врач unless you are addressing him or her directly:

Доктор, я умру? (not Врач, я умру?)