I'm just beginning to play this game but I'm a bit confused: do I have any tactical advantage with the pay version? Do I get more champions, more skills? Or it's just a matter of cosmetic stuff?
League of Legends is free to play game.
That means that if you bought it (without any additional in game stuff) you have been scammed. For example if you buy Collector's_Edition you will get champions (which could be bought with in game money got by just playing) - the more you have, the more you can try out and that's makes you a better player, but every week you can choose free champions, so this is not a big advantage. runes - which you can use it only in lvl 20 or higher, but if you are 20 lvl, you can buy them form IP got from just playing. And Riot Points which is good for buying another stuff in game.
On the other hand there is many stuff you can buy in game. But most of the stuff you buy in game (champions, runes, boosts) you can get by just playing and collecting IP. Except skins and themes which doesn't add any tactical advantage, only looks.
There really is no point to purchasing a copy of League of Legends. In all honesty, I didn't even know you could.
The free-to-play client is identical. The game itself is identical. The only things included in the physical copy (beta access [which is worthless now] and champions) can be purchased in the free-to-play downloadable client.
The champions in the physical copy can be purchased under "Bundles" in the store page, and (if memory serves) includes the runes and Riot Points included with the physical copy.