Generally, you'll suffer a pretty severe penalty if you use tools or weapons above your rank. If you're even 1 level below a tool/weapon, you can suffer a 20% penalty to the stats it offers. The farther from the weapon/tool's optimal rank you are, the more severe penalty you suffer. And the tool/weapon will become damaged much more quickly, so you'll have to spend more money on repairs.

There's a simple way to tell if the tool/weapon gives you better stats than the one you've got right now. Open your main menu, and click on "Gear." Then equip the weathered pickaxe, and check your gathering/perception/output stats (they'll show there in the numbers on the left of your screen). Then equip the Plumed Bronze Pickaxe, and check the same stats.

The Plumed Bronze Pickaxe is level 12, and you're currently a level 13 miner. So you should be fine.

But the Steel Round Knife is level 47. You're currently level 10. So I suspect you're going to be severely gimped using that tool until at you're at least level 42+.

I would suggest that you use tools/weapons that are at or below your current level, or at least no more than 1 or 2 levels above you. If you're trying to save money, you might be able to get away with buying a weapon/tool that's a few levels above you, but as a general rule, you're going to be gimped using tools/weapons above your rank. It's not so bad at the lowest levels, because you move through those levels so quickly. But you're not going to perform as well with the tool/weapon above your level until you level up to it.

The penalty isn't as severe for armor as it is for tools/weapons. So if you're going to save money, I'd suggest buying gear that's above your level, rather than tools/weapons.