True is an adjective Truth is a noun

"That's true..." is commonly used. I have never heard anybody use: "That's truth..." You can say "That's the truth..." - referring to something very specific that somebody has said...

Seeing as you meant to say 'that's the truth', not 'that's truth': 'that's true' - This is usually used to agree with someone, tell them they've made a good point, and that what they have said is really the case. 'that's the truth' - This is usually used in a more serious context, maybe when someone is giving an excuse. It feels like the person is emphasising that they are not lying about something. The adjective 'true' often feels more trivial than 'the truth', which is a noun.