Regarding QaH and boQ – There's a good deal of overlap, and either one can be used in many (maybe most) situations. The main difference is really one of connotation. In general, QaH would be used in situations where the person needing help can't do whatever needs to be done without assistance of some kind, while boQ would be used in situations where the person might be able to do the task alone, but it's easier or better or more efficient if someone helps out.

The object of boQ (the recipient of the aid) retains responsibility for the activity being aided. The person boQ-ing is secondary (though not necessarily unimportant!).

The object of QaH can't do whatever it is alone, so responsibility falls to the person QaH-ing (or responsibility is shared).

Someone who falls into a river and is unable to swim would not shout out HIboQ!.

Someone who can't reach the top shelf would probably ask another person for boQ to get something down from the shelf (even though the task could be done alone if the person used a stepladder or stool or something). If the person tried it alone and boxes toppled down from the top shelf onto the person, immobilizing him/her, you'd probably hear the buried person ask for QaH.

It would be odd to use QaHwI' to refer to an administrative aide; that's a boQ.

A parent probably wouldn't QaH a child with a homework assignment.

There is nothing weak about asking for QaH (when help is warranted), but an offer to QaH in a situation where you'd expect to hear an offer to boQ may sometimes be mildly insulting, implying that the offerer questions the offeree's ability to do something.

Having said all of that... The distinction is not always clear nor does it always matter, which is why the words are mostly interchangeable.

"Assist" would probably be translated as boQ most of the time. Other than in a few contexts, both "help" and "aid" could be translated as either boQ or QaH.

David's boQwI' app is aptly named.

(Message from Marc Okrand to Jeremy Cowan on Facebook of October 10, 2017, [archived on qepHom.de])