Underhill, Florio, and according to them the Saints coaches say JW runs essentially the same offense as Drew, but he targets the intermediate and deep stuff more often because he has the arm to make good throws downfield later.
Jameis maybe will need to check down more because of how great Kamara is, but they are not going to be installing a run and shoot offense if he wins the job.
Taysom will also run basically the same offense, but more RPO and QB runs, plus he only reads half the field and needs more time to set and throw downfield. The offensive shell, route spacing, line calls, etc will essentially be the same for both even if play calls, the way they read defenses before and after the snap, and number of scramble plays is different.
If Jameis is half as good in a Sean Payton offense as I think he can be he will win the job hands down, but there will certainly be a legit competition. Hopefully Saints fans will accept the outcome.