For hỗ trợ, the NAVER Korean-Vietnamese dictionary gives an etymology of 互助 for the verb and 相助의 as the meaning of the adjective. In both cases, the sense of it is mutual support or to help each other. I'll omit the Korean commentary since it is not relevant.
Ủng hộ’s etymology is given as 雍護 whose individual characters mean marsh/pool/harmonious and protect/guard/defend/shelter respectively. The Korean dictionary gives a definition which I can describe as supporting from behind (支持하다) or to defend by taking a side and/or providing assistance (擁護하다).
You can look them up yourself on NAVER, but the difference is basically the mutuality of the help/support (help someone else versus help each other).
I assume they're different meanings of support (physical, emotional, financial, etc) but I still don't have an answer as to what the other one specifically meant by "support" either. Would love if someone could clarify this.
What does the "ư" at the end mean? When I hover over it is says this word is untranslated.
I think it means something like putting an '!?' at the end of the sentence.
The particle ừ at the end of a sentence changes it from a statement into a question, like in English where we put yes or yeh at the end of a sentence to form a question. It is commonly used when you expect the person answering to agree with your question. To use this example: "You support me." That is a statement. You support me, yes/yeh? That is a question and you are expecting an answer that agrees with what you have asked. If you are not sure what the answer will be or less confident the answer will be yes then replace ừ with phải không. Which is like adding "yes or no?" or more commonly "right?" at the end of a statement to form a question. Ù can also be the response which would mean "Yes/Yeh" as in "Yes/Yeh I support you." As for putting không at the end of a sentence to form a question do not forget to place có before the predicate or in this instance the verb. Bạn có hỗ trợ tôi không? which forms the English question "Do you support me?"
it's similar to when you add không to the end of a question. it signifies that it is a yes or no question.
It did not accept không with me. Such differences should be introduced before. Btw I had no ư. In my sentence was hả...
as I understand it if you use khong you expect an answer but ư has some kind of rhetorical aspect, you do not necessarily expect an answer to the question.
From what other people is answering, use ư when expecting a positive answer, just as confirmation, as if you said: "you support me, right?" Even if you don't get an answer you assume it's "yes".
Or either I could make the question more explicitly:
"Do you support me?" if I am less secure about your support.