Quite different, with some overlap! I'll try and explain simply.
1. International arbitration. This is generally split into investor/state and commercial.
(a) commercial. A private dispute resolution between two commercial parties, based on a contract. No PIL
(b) Investor - state. Disputes based on treaties rather than contracts, with consideration of international law in treaties and the law of investment. This is where the overlap is. Parties are investors and states.
2. Public international law. A broad practice area, normally involving disputes between states and areas such as international human rights. Can include international supply chain issues, cases before the ECHR, as well as cases before the ICJ / ICC.
I hope this helps! I should say that the distinction between the two is not iron-clad, and there is some debate as to where the boundaries lie.