If a new season starts on each server at a different time, with differences being in weeks, the servers with the old season running will feel like they are left behind, not having access to new content, heroes, improvements introduced in the new season etc. Moreover, content produced by content creators will not be relatable.
The worst part that would happen is for new players, which will have their season cut short. Being such a player (I only play for about 10 days), I don't see this as a downside, since with season reset all my heroes will be reset to lvl 1 anyway. I would feel much worse being stuck playing Season 1 for 6 more weeks after the other servers have already started Season 2.