Your public profile is a single page that gathers all your public wishlists in one place, so people can browse everything you've chosen to make public. Sharing one wishlist sends a direct link to just that list — and that link works even for wishlists that don't appear on your profile.
The difference
Your public profile is a page at /@your-handle showing your identity — name, bio, avatar, social links — plus your public wishlists, and people find it by your handle or by following you. Sharing one wishlist is narrower: you copy that wishlist's link and send it to specific people, who open just that list with no need to visit your profile.
Which wishlists show where
A wishlist set to Public appears on your profile and opens from its direct link.
A wishlist set to Anyone with the link does not appear on your profile, but anyone with its link can still open it. It's a good middle ground — shareable, but not listed for everyone.
A wishlist set to Invite only appears nowhere public. Only you and people you invite by email can open it.
Keep in mind
Sharing a link doesn't add a wishlist to your profile. A link-only or private wishlist stays off your profile no matter how many people you send the link to. Only
publicwishlists are listed there.Pick the right visibility for the situation. Use Public for lists you're happy to show anyone, Anyone with the link for lists you share directly but don't want listed, and Invite only for lists only invited people should see. See "What is my public Wishpicks profile, and how do I set or change my profile link?".
Neither route is about editing. Your profile and a shared link both let people look at a wishlist, not change it — to let someone help manage a list, invite them by email and pick an access level (Viewer, Contributor, or Full access).