pathfinder 1e – What are the benefits of being able to remove somatic components from casting a spell?

Spells (…) might fail if you’re wearing armor while casting a spell with somatic components.

(Emphasis mine.)

The first and foremost reason to remove somatic components from spellcasting, which videogames got right, is that arcane spell failure is caused by your armor limiting your movements and getting in the way of spellcasting.

Remove the need to move and you won’t fail your spell due to your armor.

…if you’re unable to move

To cast a spell, you must be able to (…) gesture (if it has a somatic component)

If you’re somehow not able to move, you can’t gesture and the only spells you can cast are those without a somatic component.

Hold person or anything that imposes the Paralyzed condition make you unable to move, as does being Pinned, as the linked paragraph explicitly reminds.

Keep in mind that some spells require you to do some specific movement (like fanning your hands in front of you or pointing a finger to the target) that might or might not be part of the somatic components, so maybe don’t try to still those spells because they might not work anyway.

Spells that require you to look a certain way—such as all detect something spells—will work if stilled, but you won’t be able to choose which direction you’re looking at.

…if you’re polymorphed into something that can’t gesture

(…) While in such a form, you (…) and can only cast spells with somatic (…) components if the form you choose has the capability to make such movements

You must have at least one hand free to provide a somatic component.

If your hands are full, you can only cast spells that don’t have somatic (nor material) components. Some class features might give you ways to work around that, as is the case for the Magus.

Last but not least, if you’re trying to be incospicuous, there’s nothing worse than making arcane gestures. Looking at the somatic components of a spell is one of the ways to recognize it (it allows a Spellcraft check) and if you also manage to make the spell silent, nobody will ever know you’ve been casting a spell at all unless if has some visual manifestation that can be traced back to you like a cone or a ray or somebody notices your material components vanishing into thin air.

Charm Person spells cast with everybody noticing are not as effective.