What would be the difference between the two? Specifically what would cause regression model uncertainty over parameter estimate uncertainty?

Modeling (regression equations) data is inherently a flawed process. There is an uncertainty that the whole model is correct (or at least the most correct possible), and also the uncertainty of a specific coefficient estimate. This is why there is a lot of indicators that lead us to determine if a model is correctly specified, fulfills all the assumptions, estimates are consistent and efficient, etc.

Do not bother too much about those concepts, they are not that much important for the exam.

Hope this helps!