It's more damage. You do full normal weapon damage, plus int.
1d8+dex+int (10.5) > 1d10 firebolt (5.5) *assuming 3 int/dex
3d8+dex+int (20.5) > 2d10 firebolt (11) *assuming 4 int/3 dex
5d8+dex+int (30.5)> 3d10 firebolt (16.5) *assuming 4 int/4 dex
7d8+dex+int (40.5) > 4d10 firebolt (22) *assuming 5 int/4 dex.
That's close to double the damage for being in melee. Though the damage is spread out and more MAD, so call it 50% more.
That's enough.
Multi-attack on the blade singer is basically a throw away feature. Only useful if you end up in a silence.
issues I have with this comparison:
1) It assumes the secondary effect is always triggered (my biggest issue)
2) It does not take "to hit" into account (the effect of MAD is seen here right from 4th level)
3) This is a very small one, but technically the FB benefits more from crits, which also increases its damage
I suspect if all those were taken into account, GFB would still do more damage, but the difference would be a lot smaller. It still might be 50% at some levels, I didn't do the # crunching.
Primarily I would question whether GFB doing more damage than Flame Bolt makes melee and effective primary option for Wizards. I would be more interested to match a Wizard using GFB up against another class using GFB, or using their standard melee attacks. Then how does it compare?
Let's look at a Rogue. Take a rogue (race unimportant, subclass unimportant). push him up to 17th level, will certainly have 20 dex by then. Zero other optimizations. Give him a rapier in main hand and a short sword in off hand:
Main hand rapier base damage (9.5 x .6) + off hand shorstword damage (3.5 x .6) + sneak attack damage with improved chance of connecting (35 x (1 -(0.4x0.4)) + shortsword crit chance added damage (3.5 x 0.05) + rapier crit chance added damage (4.5 x 0.05) + sneak attack potential crit damage modified for extra crit chance (35 x (1-(0.95 x 0.95)) = 5.7 + 2.1 + 29.4 + 0.175 + 0.225 + 3.1425 =
40.7425
Now lets assume the Bladesinger is 17th level as indicated above. Let's modify damage by "to hit" and crit potential. We will assume the secondary effect is triggered:
7d8+dex+int (40.5 x 0.55)+(18.0 x 0.05)=
23.175
We of course know that there are ways to boost both these damages, and we both know that neither of these builds are particularly damage-intensive (a Fighter with two handed weapon for example will wipe the floor with either of these two), but a rogue at least can contribute effectively in melee, even if they aren't particularly good at it. The Wizard using GFB, not so hot. If your Wizard is a Bladesinger, I should note you should add another 3 (26.175) because of damage bonus gained at level 14. Still, really not good by comparison.
Flame Bolt is really just something a Wizard does when they have nothing better to do. Melee attacks is exactly what a Bladesinger is supposed to be doing, and they just aren't very good at it.