Your profile is one page at /@your-handle that gathers all your Public wishlists, so someone can browse everything you've chosen to show. Sharing a single wishlist is narrower: you copy that list's link and send it to the people you want, and the link works even when the list isn't on your profile.
Which wishlists show where
A wishlist set to Public appears on your profile and opens from its own link. Anyone with the link doesn't get listed on your profile, but anybody holding the link can still open it. That's the middle ground, and it's what you want for lists you hand out yourself. Invite only shows up nowhere public: just you and the people you invite by email.
Good to know
Sending someone a link doesn't add that wishlist to your profile. A link-only or invite-only list stays off the profile no matter how many people you send it to. Only Public lists get listed there. See "What is my public Wishpicks profile, and how do I set or change my profile link?".
Neither route lets anyone edit. Your profile and a shared link both give people a look at a wishlist, not the keys to it. If you want someone helping you manage a list, invite them by email and pick their access level: Viewer, Contributor, or Full access.