Why does a hydride like NaH act as a base, but a hydride like NaBH4 act as a nucleophile? A TA talked about this today in the context of ketones.

If you stick an acetone and $\ce{NaBH4}$ together in a protic solvent, you get 2-propanol.

If you stick acetone and $\ce{NaH}$ together, you get an enolate. What's the difference? They're both hydrides.
First of all, borohydride as a group is not same as hydride. So the mechanisms are very different.

Because a simple $\ce{H-}$ anion is more basic than an enolate $\ce{C=C-O^{-}}$(product of $\alpha$ hydrogen deprotonation of a ketone), it just deprotonate before reduction will happen. Deprotonated ketones (enolates) will no longer be a substrate for reduction.

Borohydride $\ce{BH4-}$ is not as basic as enolate, so deprotonation hardly happen. Thus a reduction will be the predominant reaction.