Both let someone edit your wishlist. The difference is reach: Full access can also rename the wishlist, change its settings, and delete it, while Contributor can only add and manage wishes — and a Contributor can only change the wishes they added themselves.
What each one can do
Full access — everything a Contributor can do, plus rename the list, change its settings, and delete it. It's the closest thing to a co-owner. Give this to someone you fully trust to run the list with you.
Contributor — add wishes, and edit or remove the wishes they added themselves. A Contributor can't rename the wishlist, change its settings, or delete it, and can't touch wishes someone else added.
For comparison, the other two levels sit at either end. Owner is you, the creator, with full control over everything including who else has access. A Viewer has read-only access as one invited person — they can look at the list and reserve a gift (if you allow reservations), but can't change anything. Higher levels include everything the lower ones can do.
Which one to pick
Choose Contributor for someone who should help fill the list — a friend adding gift ideas, a partner co-planning an event — but shouldn't be able to rename or delete it. Choose Full access for a genuine co-owner who should manage the whole list, settings included, the same as you.
If you're not sure, start with Contributor. You can always change it later — see "How do I remove someone's access or change their permission level?"
A few clarifications
When you invite someone, you pick from these same labels in the app: Full Access, Contributor, and Viewer. One naming note: "Contributor" here means a co-editor, not a money contributor. Reserving and gifting never involve splitting a cost on Wishpicks.
Access level and wishlist visibility are separate settings. Visibility controls who can view the list; the access level controls who can edit it. A public list can still have editors, and a private one can too.
Need someone to just view and maybe reserve a gift, not edit? See "What is the difference between viewing and editing access?" Setting up co-editing for the first time? See "How do I invite another person to edit my wishlist?"