Research subsidies capture the positive externalities to society that result from conducting that research. In other words: because much or all of the value of discovering--for instance--a new, accurate model of mental illness is leaky and impossible to capture, it's underprovisioned by the market. Government funding correct the inefficiency that would otherwise result.

That's the obvious reasoning, and I'm sure you knew this when you asked the question. Whether a specific topic has those particular externalities or not is always up for debate.