The biggest difference is that it's easier to say "у" before the б-sound. You can liken this to a/an in English, though it's more flexible in Ukrainian. There are specific rules on when to use which sound, but Ukrainians don't follow them in everyday speech.
I heard only 'basenj', couldn't hear the last syllable 'ni'. (reported)
They translate to one another and are distant cognates. A closer cognate is the English "basin" or French "bassin".