So I see many people thinking this is a Windows Phone issue, when it's actually a facebook issue (well it's a bit of both).
Basically if you go to facebook on your computer, then click on photos, then albums, you will get a thumbnail view of all your albums.
One of the first albums should be the "Profile Pictures" Album.
Now historically, whatever your most current profile picture is became the album cover for the Profile Pictures album, BUT a few months ago, facebook made some sort of change for EVERYONE, and now everyone's album cover for the Profile Pictures album is profile picture that they had 2 months ago. Make sense?
So the Me Tile picture you should be seeing (the one that's old and hasn't updated) is the picture that's the album cover of your profile picture album.
Now the annoying part is that the Profile Pictures album is the only album where you can't manually select an album cover. The way to get around that is to open that album, hover over your latest profile picture, and on the top left corner click the "Edit or Remove" Button and then click on "Make Profile Picture". There should be an message that pops up and says:
"You've used this photo as your profile picture before. Do you want to use it again?"
Just go ahead and click confirm and facebook will remake this your profile picture and this will appear as a new notification on your timeline. This action will now make your current profile picture the new album cover of the Profile Pictures album and now your Me Tile will update to your proper profile pic.
If you want any of your contacts in the People app to have the proper profile pic showing then you need to get them to manually make their current profile pic their new profile pic throught the Profile Pictures album (just like you just did). It's a ***** but for now it's the only way.
TL;DR:
Open your "Profile Pictures" Album on facebook on your computer.
Click "Edit or Remove" on the top left corner of your most recent profile picture.
Then click "Make Profile Picture"
Then click "Confirm" when a facebook message pops up.
This will update your Me Tile.