Comparing confidence intervals: CI of mean vs CI of regression coefficient


I have longitudinal data with measurements at two timepoints. I am interested in the change per year and calculated this in two ways:

* Dividing the difference between the measurements by the time difference and averaging over all cases.
* Performing a linear regression with subject age at timepoint 0.

I have 95% confidence intervals of both. The first one it is a confidence interval of the mean and the second is a confidence interval of the regression coefficient.

Can I compare these two confidence intervals? Can I conclude that if they don't overlap, the two computed changes per year differ significantly?

Thanks for you help.