Pickups (or collectible Items) are all the things you can pick up in the Tomb Raider Games, and put in the Back Pack (the Inventory)). Weapons, First Aid Items, Keys, Tools and Artefacts can also be Pickups.
Lara can pick up things by standing (or swimming) above them and pressing the Action Button.
The Gear consists of items that are (after collecting them) constantly in her inventory, like weapons and first aid items. Please not that not all gear belongs to the pickups. Even Lara's Outfits can be considered gear.
During her adventures, Lara finds a lot of different items that don't belong to her constant gear, like keys she uses to open doors and other itmes she only uses once and then leaves behind.