Upgrades for most tools do increase damage dealt to resource nodes. It's especially important for pickaxes, but also for chopping axes. The difference between base iron pickaxe and fully upgraded one is +50% more pickaxe damage, for example. But whether this is worthwhile investment or not depends.

Scrap piles for example have low durability/HP, so fully upgrading iron picks for that purpose would be a waste: It'd be better to get more pickaxes instead. Silver veins, ancient giant armor and swords, and black marble on the other hand are much more durable, but it may be more desirable to move on and progress until you get the blackmetal pick and upgrade that instead.

Like most tools and weapons, pickaxes have cleave penalty, so their damage is reduced when striking multiple chunks/nodes at the same time. Whether or not the pickaxe damage increase (together with your skill level effects) helps depends on how many objects you strike at once.