You can use a theorem without "running" it, though. The classification of finite simple groups took thousands of pages to prove; it would be pretty inconvenient if you had to read all those pages to use the theorem!

Subjectively, the distinction is useful. I'm never going to have a different brain, and I won't outlive my brain, so for all intents and purposes, my brain is an eternal constant. Theorems that ran on my brain once have accomplished their goal of convincing and never need to be run again. They are immutable.