It is not so easy to state a clear distinction between the A- and B-theories of time. One says things like: "According to the A-theory, there is an objective past, present and future". But although I've used formulations like this, I've cringed when doing so, because of various problems.
So, let me propose a different characterization: The A-theorist holds that at different times we inhabit different possible worlds, while the B-theorist denies this.
This formulation makes the A-theory seem implausible--according to the A-theory, interworld travel is possible, since if we wait, we will end up in a different world from the one we're in. Nonetheless, I think this formulation is right, follows from the "objective past, present and future" formulation insofar as we can understand it, and the A-theorist should agree with it. Consider that a world includes or encodes everything that is objectively the case, and if something is objectively the case at w1 but not at w2, then w1 and w2 are different worlds. According to the A-theorist, the difference between some event's being future or past is objective. So, if we consider the world we'll be in in ten minutes, there will be objective differences--some events that now are future will then be past. Hence, it's a different world.
Another reason the A-theorist should embrace this is that it allows one to neatly characterize the difference between A-theories with an open future and A-theories without an open future. On A-theories with an open future, for a future time t, there is a minimal set W(t) of worlds such that it is a fact that the world at t will be one of the members of W(t), and typically W(t) has more than one member. On A-theories without an open future, for any future time t, there is a unique world wt such that it is a fact that the world at t will be wt.
There is, however, an alternative to the above approach for the A-theorist. The argument for my characterization of the A-theory depended on the claim that possible worlds do not change. The A-theorist could deny that. But I think it would be an implausible denial. Possiible worlds are like propositions. They are abstracta, not spatiotemporal entities. They do not change. There is change in a possible world, just as there is change in a novel (i.e., in what the world or novel describes), but the world itself does not change, just as a novel itself need not change (it might, if it gets revised or destroyed, but that's a dififerent change from the change in the novel).