It depends on the spell

The exact wording of the spell will tell you when it does damage and if it is the same damage or needs to be rerolled.

For example:

• Heat Metal is not rerolled: "... you can use a bonus action on each of your subsequent turns to cause this damage again."
• Spirit Guardians is rerolled each time there is a saving throw: "On a failed save, the creature takes 3d8 radiant damage ..."

It depends on the spell

The exact wording of the spell will tell you when it does damage and if it is the same damage or needs to be rerolled.

For example:

• Heat Metal is not rerolled: "... you can use a bonus action on each of your subsequent turns to cause this damage again."
• Spirit Guardians is rerolled each time there is a saving throw: "On a failed save, the creature takes 3d8 radiant damage ..."

That said, most spells do call for rerolls and it wouldn’t make a big difference if you played all spells that way. I would be cautious about going the other way since refilling reduces the damage variance and lower variance benefits the PCs over the monsters (since the PCs use and are exposed to many more damaging spells than any given monster).