Silenced creatures will lose their turn if they try to cast spells. Party members may not select the Magic, Item, or Drink command when Silenced. Silence wears off at the end of combat. It can also be cured with the AMUT or CUR4spells.
Spells a creature was already casting will be interrupted, but creatures may not choose to cast spells in later rounds if they are Muted. Mute wears off at the end of combat. It can also be cured with the Vox or Cure4spells.
Amnesia is the permanent version of Silence, preventing a creature from casting any spells. On your front line fighters, this is usually not a problem, except for the permanent status ailment restriction on after-battle progress. It takes an Esunaspell of level 4 to cure Amnesia during battle, or Esuna 3 outside of battle.
Silence simply stops a creature from casting spells. If the creature is Silenced before their turn comes up when they would cast a spell, the spell is interrupted. You can't choose to cast a spell at all if your turn starts when Silenced. Like all temporary status ailments, it can wear off at the end of any turn. It can be cured manually with Basuna 2.
Creatures with Mute cannot use spells or Lores. They will still try if using Rage or Sketch or the command was set before Mute was inflicted, but this will fail for any ability with an MP cost.