"Scientist" is used when referring specifically to a person who is an expert in a science, especially physical or natural sciences.
"Scholar" is more broad, and can be used generally for anyone who has profound knowledge of a particular subject.
"Scholar" isn't used for sciences (chemistry, geography), but can be used for History, Language, etc. Examples of this are:
Language scholar - Yes.
Psychology scholar - Wrong.(Psychology is a science)
Greek scholar - Correct.
Thus, "scholar" is only used for someone who is an expert at a particular subject, when the subject is not a science.
An additional meaning of "scholar" is:
a student who has been awarded a scholarship.
Thus, yes, "scholar" can also be used when referring to students who have received scholarships, as you asked.
Your second question below:
A "researcher" implies someone who is doing scientific research specifically as a vocation e.g. I am a researcher at Harvard University. If someone like a journalist was just doing research for an article, he wouldn't be referred to as a researcher, but rather as "A journalist researching".
A "researcher" could be a scientist, or a scholar. A Biblical scholar could be a researcher at a seminary. A chemist could be a researcher at a pharmacist. But when referring to the person, it doesn't matter if he/she is a scientist or a scholar, just use "researcher".