"strainmut.minute15" is the difference between Mut vs WT at minute 15, controlling for baseline. If you add "strainmutvs_wt" to this, you get the LFC for Mutant vs WT at minute 15, not controlling for baseline. So the second one is the observed difference at minute 15 between the two groups (because you added in the change that was present at time=0).
I have seen this answer and similar in different posts.
What I would like to better understand is the highlighted part.
What is the exactly difference between
The first is the difference between the strains at minute 15, controlling for the difference present in the strains at the first time point (t0). That is what we mean by "controlling at baseline".
The second is the sum of the baseline difference and the difference between the strains at minute 15. If you think about four groups of samples:
A: wt at t0
B: mut at t0
C: wt at t1
D mut at t1,
Then the above is giving you (B-A) + ((D-C) - (B-A)). Which gives you D-C, i.e. the difference between Mutant and WT at minute 15, not controlling for baseline. Typically, you want to control for differences at t0 though. I should probably clarify that post to make that more clear. The person was asking for a strange type of analysis, and I gave the answer, but that is not recommended or the usual way to analyze time series data.