The short version: viewing lets someone look at your wishlist and reserve a gift, while editing lets an invited person actually add and change the wishes. Viewing usually works from a shared link with no account; editing requires an email invite and an access level.
The access levels
Owner — you created the wishlist; full control over everything.
Full Access — can do everything a Contributor can, plus rename the list, change its settings, and delete it (like a co-owner).
Contributor — can add, edit, and manage wishes, but can't rename or delete the wishlist. A Contributor can only change the wishes they added themselves.
Viewer — read-only access for a specific invited person. They can look at the list and reserve a gift (if you allow reservations), but cannot change anything.
When you invite someone, these are the exact labels you'll see to choose from: Full Access, Contributor, and Viewer (Owner is just you, the creator).
Anyone you simply send the link to (a "guest") gets the same read-only experience as a Viewer — they can look and reserve, but not edit.
Viewing vs editing at a glance
Viewing — look at the wishlist, see the wishes, and reserve a gift if enabled. No invite needed for a list shared by link.
Editing — add, edit, and manage wishes. Requires an email invite and a Contributor or Full Access level.
Keep in mind
Editing access does not depend on visibility. A wishlist shared by link can still have invited editors, and a private one can too — visibility controls who can view, while the access level controls who can edit. A Viewer can never edit, no matter how the list is shared. If you want someone to change wishes, invite them as a Contributor or Full Access.
Reserving a gift is something a viewer or guest does. It isn't editing, and it doesn't change the wishlist's contents. (For how reservations work, see the Gift Reservation articles.)
To grant edit access, see "How do I invite another person to edit my wishlist?" If someone you invited still can't edit, see "Why can't someone edit my wishlist?"