My 2 cents:
They're the same thing, both of them do exactly the same thing... But...
Oftentimes when someone says that they made a balanced modulator, it doesn't include an internal CV oscillator, while a ring modulator does. You'll notice on a lot of commercial ring mods theres a CV in, qualifying it for both categories... But actually, some modular synth ring mods have no knobs, just 3 jacks: Input, CV in, and out.
They are exactly the same thing... different words meaning the same thing applied to similiar devices, I guess.
Oh, and "mod" means "modulator", so when someone says "ring mod", they actually mean "modulator".