The freedom of movement spell only prevents difficult terrain from affecting our movement. The saving throw happens regardless. The effects of failing the save depend on whether you think the terrain is what forces us to make the save in the first place
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 our movement. Note: it does not provide immunity to other effects of difficult terrain, such as changes to our speed, hit points, or anything else.
Spells and magic effects cannot reduce our speed.
Spells and magic effects cannot make us paralyzed nor restrained.
To further support that freedom of movement only applies to the movement affecting effects of difficult terrain: there are also spells like spike growth, which create a damaging area of difficult terrain; freedom of movement will not somehow prevent us from taking damage; it only prevents the area from affecting our movement.
The Gibbering Mouther's Aberrant Ground feature is not magical, so the latter two points do not apply. What the Gibbering Mouther's 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 in that area (a 10-foot radius), it must make a saving throw; failing this saving throw makes their speed become 0.
I view these as separate effects. The area is difficult terrain, and a creature that starts its turn in the area must make a saving throw. The feature never says that being immune to the normal effect of difficult terrain makes us automatically succeed on (or not have to make) the saving throw. The feature never says that certain creatures are immune to the save, so any creature that starts its turn in the area will make the saving throw.
The terrain is still considered difficult terrain, and freedom of movement only prevents difficult terrain from affecting our movement. Simply making a saving throw is not affecting our movement so we would make the save no matter what. If we were to fail the save, there are two possibilities:
Failing the save does not count as difficult terrain affecting our movement, instead the Gibbering Mouther or the Aberrant Ground feature itself is affecting our speed; our speed, and thus movement, will both reduce to zero.
Failing the save does count as the difficult terrain affecting our movement through affecting our speed; our speed will reduce to zero but our movement will remain the same. This is a strange, though not inherently impossible, situation.
In the end, we will make the save no matter what as making a save has no effect on our movement and freedom of movement only prevents difficult terrain from affecting our movement, nothing else. That said, what happens when we fail the saving throw depends on whether a GM believes the difficult terrain is what caused us to make the save.
Nothing in the description links the terrain to the save; however, the feature is called "Aberrant Ground" so I can see arguments made both ways.
Personally, I believe the difficult terrain is not forcing us to make the save; the two effects of Aberrant Ground are separate, neither relies upon the other. The creature makes nearby terrain difficult and forces nearby creatures to make a saving throw. Additionally, nothing says that the terrain itself is forcing us to make the saving throw. This also prevents the odd scenario of having full movement but zero speed. And finally the two effects of Aberrant Ground occur at different times, difficult terrain is always difficult, but our speed can only be reduced at the start of our turn. This leads me to believe the effects are entirely separate and that the difficult terrain is not what causes us to make the saving throw.