Given that mathematics and reality are always going to have the fundamental difference of ideal vs actual, every possible mathematical description is equally dissimilar to reality in this regard. However, the fact that mathematical models can accurately describe reality (i.e. the "unreasonable effectiveness of mathematics" ) implies that the ideal is instantiated in the actual ... so this dissimilarity diminishes as those models approach greater accuracy.An analogy is (1) a similarity between two things that are otherwise dissimilar, and (2) a comparison based on such a similarity.
So as mathematical models increase in accuracy, we're talking less and less about "two similar things that are otherwise dissimilar," i.e. an analogy, and more and more about the very structure of physical reality.
We're not merely making comparisons between models and reality, we're also making predictions of new phenomena and connections between previously known phenomena on the basis of these models. If the models were only connected to reality by way of comparison, this would not be possible. It is only possible in virtue of the model actually capturing/elucidating/explaining a feature of reality.
Explanation is not comparison. Explanation is not the recognition of similarities between things that are otherwise dissimilar. Explanation is a deepening understanding of reality in the sense of understanding the very organization principles by which it is comprised and by which it evolves into new states. Again, if these underlying principles were not accurately grasped (an act of comprehension well beyond the scope of any mere comparison), then building upon that understanding (e.g. predictions and technology) would not be possible.
For example: I can make an analogy between a car and a cart. They are two things with wheels that can move about (and in every other sense dissimilar). However, this comparison in no way elucidates the nature of the automobile. It does not make it possible to explain how cars move on their own power, much less how to reproduce that technology yourself.
See what I mean?
What you guys are doing is making an analogy between scientific explanations and analogies, and then confusing this former analogy for the latter. It's circular reasoning.*
*[edit: to be fair to WF, he did say that models are like analogies. I agree that one can draw certain parallels between the two concepts, but--as I understand WF's apparent meaning--similarities don't necessarily imply an identity between two things. So, I'll accept that mathematical models are superficially like analogies, but in their most important roles, they are nothing at all like analogies.]