Having everything in one launcher is nice, but the big advantage is that it will try to enable the steam in-game interface.
This isn't always a good thing - you may want to disable it for some games if it makes them unstable.
Launching non-steam games from steam will also display to your friends that you are playing that game.
For me, this is a critical feature.
It will not track your play time on that game, however.
Update: You can now in-home stream non-steam games if you add them.
Thanks to the other replies for mentioning this.
gaming.stackexchange.com/users/1895/robertpateii
The advantages (already mentioned) are that you can have all game shortcuts in one place, other people can see you playing that game and you have Steam overlay in-game.
There are also some disadvantages.
If you are downloading or updating a game on Steam launching the non-Steam game stops the download/update; this is the default behavior of Steam.
Useful if you are playing a multiplayer game, but annoying if you are playing a single player game.
But the main disadvantage is that not all games are compatible with Steam overlay.
Just as example I can cite League Of Legends.
If you try to chat during game the chat stoles the focus and you game does not get anymore key inputs; the only workaround in this case is Alt+Tab to desktop and than return to the game.
gaming.stackexchange.com/users/730/Drake
Non-Steam games added to Steam can be streamed with In-Home Streaming.
gaming.stackexchange.com/users/9567/Ben Jackson
This way you can let your ever eager to play friends know what you're busy doing besides games.
Sometimes I'm playing Microsoft Visual Studio 2017 , so everyone knows I'm not really up to much else at the moment.
Also, it's fun to be playing ridiculous non-game stuff.
And games are about fun, so it's a nice feature.
gaming.stackexchange.com/users/23901/user1306322
Having all your game library all in one place has the advantages of having only one launcher which can lead to a cleaner desktop.
It allows you to remember those older or indie games you don't play as often anymore.
gaming.stackexchange.com/users/1770/Elpezmuerto
Using Steam allows you to make screenshots and chat via the steam chat system (including voice comms if set up) even if the game itself doesn't allow it.
I've found the screenshot feature very useful.
gaming.stackexchange.com/users/39583/Rob
These days, it also enables Steam Gamestreaming if you add a non-steam game to your steam library so all your friends can watch in envy as you play Elite: Dangerous.
gaming.stackexchange.com/users/105098/Henry van Megen
Is there a point in adding a non-Steam game to Steam?   Are there any advantages besides being able to launch the game from within Steam?
Is there any point in adding a non-Steam game to Steam?
gaming.stackexchange.com
<steam>
gaming.stackexchange.com/questions/9616
gaming.stackexchange.com/users/330/Nifle
You can have all the game shortcuts in one place. It also enables gamestreaming if you add a non-steam game to your library. There are a variety of pros to this, chief among them being that you have all your games in one place, and this allows you to remember what you have better. It also allows you to stream the games you play, show them to your friends and make screenshots of them.
Users say that it is good to keep all games in one place. Users also say it has the advantage of allowing to stream the games. It allows players to remember their old games. Users say they find it nice that they can show their friends what they are playing It also allows to make screenshots