Biblically the biggest problem with modalism is that you end up having God talking to himself several times in the NT.
The idea of modalism is simple enough - God has different "modes" of being, kind of like an actor who simply appears with different masks in different situations . If the same person is merely appearing in multiple forms simultaneously, there are some weird situations in Scripture that result, and have implications that are dicey.
When Jesus is Baptised, a voice from heaven declares, "This is my son, with whom I am well pleased". If Jesus is just a different view of the same God, why would he say something like this, to himself? It makes God seem a bit arrogant, frankly.
When Jesus is on the cross, he cries out "Father, Why have you forsaken me?" How could he forsake himself if he is the same person forsaking and forsaken. Note - specifically in relation to the cross, the view that the Father was on the cross is a heresy known as Patripassianism - from "the Father suffering"
In Hebrews 9:14, "the blood of Christ, who through the eternal Spirit offer[s] himself unblemished to God". How can Christ offer his own blood to himself if he is the same person? Unless that were somehow possible, Substitutionary Atonement is also invalid.(See Note 2)
By maintaining the distinct personalities (and personhood) of each part of the Trinity, you avoid all of these ridiculous to downright bad implications.
1 Or kind of like the Doctor meeting his past selves, crossing the timelines, and somehow necessitating running up and down corridors a lot. But I digress!
2 Note - Substitutionary Atonement is only one theory of salvation, so even if you could some how prove Modalism true and S.A. false, it wouldn't deprive us of salvation - it would just upset a lot of people who think this legal metaphor is the only theory that describes salvation.