Bilarczyk vs Bielarczyk

Those in the know--In a village as small as Raba Wyzna in Nowy Targ, Lesser Poland--would the spelling of Bilarczyk vs Bielarczyk possibly indicate the same family simply filtered through different spelling conventions/transliterations--or is this just wishful thinking?

My immigrating ancestor spelled his name as Bilarczyk on naturalization documents, and the only folk I'm finding back home near his 1861 birth in Poland used Bielarczyk spelling instead. I've also noticed many immigrated to other places around the same time, (Chicago in Illinois, Minneapolis in Minnesota, etc), and wonder if it is likely related folk would choose such different landing places, or instead chain-migrate. Any insights welcome, thanks!