Mood stabilisers are designed to prevent moods from hurtling up and down, antidepressants are designed to lift someone out of depression.
Mood stabilisers are mostly used to “quiet” manic phases, more than to treat depression. I take lamotrigine (as a mood stabiliser) which is meant to be effective for bipolar depression though, and I take it along with a low dose of the antidepressant fluoxetine.
Antidepressants are difficult for bipolar patients as taking too high a dose can lead to mania. Which is where the usefulness of mood stabilisers comes in, they are used to prevent the acute ups and downs.
The only “true” mood stabiliser is lithium. There are a lot of drugs which are used as mood stabilisers but are actually other classes of drugs, e.g. antipsychotics, anticonvulsants, etc.
Hope that helps