A bit of background on vaccines....
It takes about two weeks (from 10 to 21 days if I recall) for the immune response to kick in upon exposure to a foreign bug. On average, the response is about 65% (this is in theory based on statistics, in practice it is different and depends on the type of vaccine). A second exposure will boost this response to about 95% (and a third to 99%, again this is on paper)
In modified live vaccines, you are injecting a live organism (that is not supposed to cause disease). In theory again, by the time the immune system kicks in (two weeks later), that orgainsm would still be around to stimulate the immune system again (it is live therefore it is still multiplying), thus negating the need to booster in most live modified vaccines.
In killed vaccines, the antigens (the part of the bug that stimulates the immune response) are used up when immune system responds (because it is dead, it does not multiply). Thus in order to get 95% coverage, you should booster 2 to 3 weeks after the 1st shot.
Each manufacturer uses their own method for making a vaccine, they do not necessarily "make" the same antigen (the part that makes the immune response quick in) even though it may be the same bug. To oversimplify, one vaccine may use the "head" of the bug, another vaccine may be using the "legs" of the bug to stimulate the defense response.
Will using another brand to booster work? The short answer is: I am not sure that would work if you looking for 95% coverage. The manufacturers cover their behinds by saying booster with the same product and they will never fund studies using a competitor's product as a booster.