I'll list the pros and cons of using both

Immortal

pros:

1. Instant Banish no fuss
2. More HP, damage (not important but is a plus), armor and 1 extra awakening slot.
3. 4 Immortal Staff mages in party can clear any higher floor fast and fewer deaths (tested by me and guildies/friends).

cons:

1. Costlier compared to Flame forged.
2. Can't banish bigger mobs.
3. Needs at least 2 mobs for the proc to work.
4. Proc unreliable sometimes (but gets the job done most of the time).
5. Proc not that good in lower floors.

Flame Forged

pros:
1. Cheap
2. Comes in 4 variations
3. 1 mob is enough for the weapon to proc but the weapon really shines against big pulls.

cons:
1. High chance to get cursed and die.
2. Low HP, armor and damage (damage not important tbh).
3. 3 awakening slots.
4. Not that good in higher floors against big pulls, takes time to kill them.
5. 4 Flame Forged Staff mages in party is quite frankly not that impressive (Tested by me and friends/guildies).

It's upto u to decide, at one point I kept both handy. I used Flame Forged from Maus 1-3 and then switch to Immortal in Maus 4-5. Later I decided to keep using Immortal and part ways with Flame Forged, never regretted.