I should also mention that if you delete a bulb (or group) from Hue, the device will remain on Hubitat. To avoid confusion, it's probably best to make a change on Hubitat whenever you make a corresponding change on Hue (if you have that device exposed to Hubitat), but you have free reign to do what you want on either system for the most part (obviously, Hubitat can't control a bulb or group whose ID no longer exists in Hue, for example--which should only happen if you removed it from Hue).
Whether you will have to change anything with the group on Hubitat depends on what you did on Hue. If you kept the group on Hue and just changed bulbs and/or changed the name of the group, the Hue group ID should remain the same (unfortunately, there's no good way to see that in the Hue app like you can for bulbs). This means the Hubitat device should still "point" to the same Hue group. You can rename it on Hubitat if you need to. If you created a totally new group on Hue, you'll need to add that new group to Hubitat (even if the name is the same, as again, Hubitat associates with Hue devices based on an internal Hue-Bridge-assigned ID for lights or groups).
While it might be nice to have a "name sync" option between the two, nothing like that currently exists. If it did, I'd personally want it to be optional. 
But I definitely understand the desire!
Side note: I think there's also a possible point of confusion where the Hue Integration app will show devices already added to Hubitat by using their Hue name instead of their Hubitat name, which may be confusing if they are different. I haven't checked that in a while and might be remembering wrong...
