I usually consider using some form of Monte-Carlo method to
investigate the distribution of the parameters of interest.
For instance your models are in reality of the form:
where r is a noise term usually (if things are going right) zero mean
normal with sd s say, which you can estimate from the residuals.
(note if the residuals show noticeable departures from independence and
normality you are in trouble and need to reconsider the models)
Now assume some value of a, b and c (the same for both samples) and
generate 100 (or a 1000 or whatever) replicates of the estimates of
(a1-a2, b1-b2, c1-c2), and compute an estimate of the covariance matrix
of this vector for the 100 replicates.
The vector (a1-a2, b1-b2, c1-c2) that you observe should be more
or less multivariate normal, and we now have an estimate of its
covariance matrix, so we can test it (you will have to look up test for
This sort of analysis is of an exploratory nature and will give a rough
indication if the models are significantly different.
For a more theoretical (academically respectable) approach you will need to
look at a reference on regression models, or general linear models (GLM).
There should be a vast literature on testing such models, which I am not