I am not going to add to the conventional list of differences between standard cosmology and the explosion picture, but I will say a few things about the general problem with these kinds of discussions in the public.
The first fundamental problem with the question is that it contains at least two fallacies. One is the false dichotomy fallacy that there are only two models and that therefor one of them has to be right and the other one has to be wrong. That is completely false. We simply do not know what the universe really looks like outside of the observable universe, hence any statement about its shape is local and therefor incomplete. If we take this limit to our knowledge seriously, then it should be obvious that we have to allow for an infinite variety of cosmological models that may or may not turn out to fall into some broad equivalence classes upon further analysis.
I don't know of anybody who has done a logically self-consistent analysis of the global situation so far (and I doubt we have nearly enough knowledge about the necessary theory, yet, to even perform such an analysis), hence I can't tell you what the result of such a broader approach would be. We can be assured, though, that it contains far more than just these two trivial cases.
In all likelihood we can assume that even the local expansion model is merely an approximation, so even if you manage to convince the layman that "expansion it is", did you really educate them about proper scientific thought or did you merely enforce an orthodoxy that stems from a rather limited view of the problem that was predominantly popular in the 1960s and maybe 1970s? I would suggest it's the latter. Personally I do not believe that this is a service either to the public or to science PR. It would be far better to educate them that cosmology is a discipline that keeps an open mind and that is aware of the approximative nature of statements about the cosmos. If we do that, then it becomes much easier to motivate active areas of research like inflation and more daring models like cyclical conformal cosmology etc..
The second major fallacy is the implicit assumption that both you and the layman understand the dynamics of explosions and that only the layman is struggling with the phenomenology of expansions. I would instead claim that neither of you know enough about the dynamics of explosions and that this lack of knowledge also applies to the majority of cosmologists. Those who do know more are most likely the ones who are also actively working on supernova physics and they happen to know that the phenomenology of real explosion is neither homogeneous nor isotropic. It is a turbulent and messy process that leaves a complicated aftermath. In other words: the cosmic explosion imagery is mostly based on a cartoon version of an explosion rather than on physically observed processes that show instabilities and asymmetries. In a serious scientific context explosion cosmology would therefor have to explain the detailed mechanism by which this special explosion would have yielded a CMB with a homogeneity of one part in 100,000 when no other explosion ever observed is even remotely that "smooth".
Incidentally this problem also exists for expansion and it is, to the best of my knowledge, still not solved satisfactorily. Inflation is one attempt to solve it, but you can find serious arguments in the literature (at least serious in my eyes as a cosmology layman) that indicate that inflation might actually be the wrong approach entirely. To me this is yet another reason to keep an open mind and to explain the problems in cosmology ("What are the observations that we have to explain?") rather than to force a toy model on the public that may or may not explain these observations well enough.
You are, by the way, not the only one who struggles with physics communication. Take Steven Weinberg, who is probably one of the smartest physicists ever. He wrote a beautiful book called "The First Three Minutes" that, to a large extent, gets most of this right... but it ultimately turned out to be wrong about the big picture completely. Weinberg preferred, if I remember correctly, a cyclical big bang model in that book. That model has been completely ruled out by now.
So what are you supposed to do? I would say, try to keep an open mind and motivate to your audience why and how you are doing that. That, if anything, is the best approach to science IMHO. We are trying to get answers rather than trying to give them.