I don't think the first spawn of it will make anyone snowball. The game starts, lets say no one goes middle to fight, and we just lane (as you should). 25 seconds after the minions spawn, the first wave of them have run into each other and you are just beginning to gain CS and poke at each other. No one will probably be down much if any HP. If your team goes from their lane to the middle point to grab it, they gain almost nothing, and left lane and lost CS for it.
However, later in the game, if you are pushing hard, and push past mid (normally a bad thing to push hard early) and you have traded a few blows back and forth, you can grab it and refresh yourself to stay in lane a little longer. Its a sort of reward for pushing hard and making yourself gankable.
Realistically, whomever clears the jungle the fastest from start will go mid to grab the relic and heal before gank, and it will boost the person. So it really effects jungling and ganking champs the most.
Later in the game, you will see champs grab it randomly on both sides when rolling in from one lane to gank another. It will probably help a few people escape ganks too, but not have a huge effect.
Teams that kite well, will enjoy it, as you can lead them behind you as you dmg them with eve/singed and hit that relic in the mid point to make sure you get out unharmed mostly.
Champs with a good leap like rengar, kha, or jax will sit in the bush waiting for people to go for it, and gank them.
Its not going to snowball the game, it isn't an advantage to get it when your health is already high, it doesn't give you an advantage unless you are already fighting, and then at that point its minor. It will create a team fight often at mid point when the 2 jungle having cleared and go to hit it. This is where you will see first blood happen (hopefully) and you will see the team with good fighting ability at level 2-3. It helps sustain people who push hard, or jungle fast.