What is the difference between "finalize" and "resolution"?

Finalizing is completing the steps of playing an item (choices, costs, legality) and turning it from Pending into a Finalized Chain Item; resolution is the later step where the item's actual game effects execute.

Finalize is Step 1 of FEPR (Rule 337): the controller makes choices, determines and pays costs, and checks legality, then the item stops being Pending (Rule 359.1). No game effects happen here, and finalizing doesn't pass priority (Rule 337.1.a).

Resolve is Step 4 (Rule 340): the newest Finalized Chain Item executes its rules text in its entirety (Rule 340.1). For Units and Gear, finalize and resolve happen back-to-back — they leave the Chain and enter the board immediately (Rules 337.2, 359.2). Spells instead linger on the Chain after finalizing so opponents get a Reaction window, and only resolve later (Rule 359.3.c–d).

The practical difference: finalize = "is this card valid, paid for, and locked in?", resolve = "what does it actually do?"