Which one is best for mood swings, lithium or tegretol?

Mood swings: Lithium would be your first choice.

The best: Would be a visit to your psychiatrist for their opinion. There is no "best", each patient should be treated as an individual.

Both can help: Both lithium and tegretol can help. If mood swings are very frequent, though, an anticonvulsant like tegretol may be more effective.

Both work: These two medications are both effective at treating mood swings that are a function of bipolar disorder. However, Lithium Carbonate has more research to suggest that it is effective for bipolar disorder and also more specifically has been shown to reduce suicidality to a greater extent than carbamazepine. Additionally, Lithium Carbonate has a greater capacity to treat bipolar depression.