I am wondering what is the difference between the two formulations.

Limit once per game appears for instance on this card and Group limit once per game appears only on this card .

I saw that discussion somewhere before. If nothing else is printed on a card limits are per investigator, so here per investigator and game, whereas the other cards can only used by ONE investigator in the whole game.

Yep, Once per game applies per investigator, so if you have 4 investigators, they can all get healed once at the hospital. Group limits mean that one investigator can do it once only during the whole scenario, so the whole group can buy 2 clues at Northside once only.