These two verbal roots *bʰewg- "flee" and *bʰegʷ- "flee" share the same meaning and very similar forms, the only difference is their ending consonant. I wonder whether they are from a same root or just a coincidence?
The reflexes in daughter languages support the reconstruction of separate roots (supported by the simultaneous existence of φεύγω and φόβος), but the semantic and phonological similarity is striking. It is probably coincidental, meaning that systematically deriving the two from a single root is beyond our technical capabilities. It's not uncommon to have multiple similar-looking roots with similar meanings, such as bhlegʷ, bhlew, bhlei, bhrew with meanings "bloat, swell": more examples here.