Facebook – What setting controls whether or not people I’m not friends with can “like” the public Facebook profile photo

Check your privacy settings: https://www.facebook.com/settings/?tab=privacy

Click on How You Connect Edit Settings. Look at the Who can post on your timeline? if it's set to Only Me this will allow people to see your posts, but not be able to do anything with it.

Edit You also cannot see the link or comment buttons when you're looking at your page as someone else.

By default, when you make an image your profile picture, it doesn't matter if its friends only or private, when its made into your profile picture, the profile picture automatically gets made public.

So if you don't want people see the full size version of your profile pic, after you setup your profile pic, you should change the privacy of the pic back to friends only, or whatever settings you like. You cannot change the privacy setting of the thumbnail profile picture however, as you used to be able to do a few years ago.