Bill Blythe had started a job in Chicago and the family was planning on relocating. So "Bill Blythe III" would have been raised in the Chicago suburbs. Assuming his mother and father remain married, that'll likely affect his own upbringing and personality in various ways, as will growing up in the suburbs of a major city rather than small-town Arkansas. And if Bill decides to go into politics in adulthood, he won't be doing so in the small, parochial politics of Arkansas. That's going to make his political career likely more difficult and drawn out, and there's of course a good chance he won't actually get into the same positions.
So, short answer: lots of big changes -- easy to pinpoint what would have been different, but who knows how it would have all shaken out. Bill himself in his memoir said he mused to Hillary that his father not died, they'd have grown up just a few miles from each other - but through the vagaries of fate, may never have met.