Why would I choose a Palladium Drill over a Palladium Pickaxe?

The wiki has this to say about drills and the latest updates:

History

• 1.2: Drills are no longer material. Each drill now has a pickaxe equivalent, for those who don't like drills.

Pickaxes are pretty much always better than their drill counterpart as of patch 1.3.0.8. All pickaxes and drills of the same ore have the same pickaxe power and mining speed (except orichalcum, where the pickaxe has a faster mining speed).

However--

Pickaxes have the advantage of having knockback (which can help with enemies when you are mining). They can also be affected by any melee weapon prefix while drills cannot. Importantly, this means that they can get the Light prefix, which increases mining speed by 15%. Other prefixes might give lower mining speed, but more damage and knockback, which might be preferable for protection when mining (and in some cases actually yields the same mining speed value due to rounding).

Drills offer only aesthetic differences over their pickaxe counterparts. They have a different profile and a different sound when used.

I recommend using the pickaxe.

All picks are faster than their drill counterparts. See this video for a comparison of mining speeds between pickaxes and drills.