I have never heard "Two different ball games" used as an expression like "Zwei paar Schuhe." Ever.
One of Rominara's links has someone called Steve UK saying it would work for him. Does that make it an idiomatic expression? I think not.
A widely used phrase is "It's a whole new ballgame!" but it doesn't mean anything like "zwei paar Schuhe."
More natural-sounding phrases would include:
Those are two different things.
Or, referring to one of the two:
That's something else entirely.
That's a whole different thing.
And there are probably more, but coming up with them would require a higher blood caffeine level than I have at the moment...
And there's also "That's a different kettle of fish," which, it should be noted, arose from "a fine/pretty kettle of fish" meaning a mess or a muddle.