What is the difference? 演讲, according to my dictionary, is either a speech (in general, like a politician would give) or a lecture (at a university or so), while 讲课 is just the lecture, as in a lesson. Is that the correct way to distinguish the two?
is strictly a verb meaning:
to teach/to give lessons (in ....)
can be a noun meaning
or a verb meaning
to give a speech
Both could be translated as: lecture, if you wanted.
The biggest difference is the 课 in 讲课 - meaning class or course.
講課 usually exclusively refers to a lecture in a school or university, while 演講 is not restricted to such context. Every kind of (somewhat formal) "speech" may be said a 演講.