The difference is in the details: both are fermentation drinks, but they are two completely different cultures used for fermentation.
Ginger bug is a batch liquid created by wild fermentation of the microbial flora of fresh ginger.
Water kefir, on the other hand, is a different culture. This is fermented by the addition of so-called water kefir crystals. The crystals are a so-called SCOBY, a symbiosis of yeasts and bacteria. They differ from ginger bug by the microorganisms they contain, the taste and the fermentation process.