Looking at the デジタル大辞泉 definition for ふらつく, it gives as a synonym ぶらつく. They seem very related, but is there any usage difference between them perhaps due to differing etymology?
I am a native speaker of Japanese but not of English so my English may not be natural but I hope to be helpful.
ぶらつく is similar to ふらつく in the following meaning. ぶらつく is to hang around. You can say 昨日暇だったから、町をぶらついていた or ふらついていた (I hung around the town as I was free yesterday.) But ふらつく expresses you have less purpose than ぶらつく.
And ふらつく can mean to stagger. So you can say 酔っ払いがふらついている (A drunk man is staggering.) ぶらつく doesn't have this meaning.