The freedom of movement spell will help against the difficult terrain, but not the saving throw; The spell only prevents your movement from being affected by difficult terrain
The freedom of movement spell states:
For the duration, the target's movement is unaffected by difficult terrain, and spells and other magical effects can neither reduce the target's speed nor cause the target to be paralyzed or restrained.
From that we can conclude what things the spell does:
Difficult terrain does not affect your movement. Note, this says nothing about your speed or anything else difficult terrain might cause.
Spells and magic effects cannot reduce your speed.
Spells and magic effects cannot make you paralyzed nor restrained.
The Gibbering Mouther's Aberrant Ground feature is not magical, so the latter two points do not apply. What the feature does do is this:
The ground in a 10-foot radius around the mouther is doughlike difficult terrain. Each creature that starts its turn in that area must succeed on a DC 10 Strength saving throw or have its speed reduced to 0 until the start of its next turn.
Ground within a 10-foot radius becomes difficult terrain.
If a creature starts its turn within a 10-foot radius, it must make a saving throw; failing this saving throw makes your speed become 0.
The feature says nothing about the creature being immune to the saving throw if they are immune to the normal effect of difficult terrain. So even with freedom of movement we will still be subject to the effects of Aberrant Ground. The ability works regardless of whether we ignore the terrain, it only requires that you be in the area.
The freedom of movement spell will not help against this ability because the spell only prevents your movement from being affected by difficult terrain.
A similar argument would be made for a spell like spike growth.
It creates a damaging area of difficult terrain, but freedom of movement will not somehow prevent you from taking damage, it only prevents the area from costing additional movement.