Jimmy White was the Ronnie O'Sullivan equivalent back then I suppose.
Back in those days, snooker was able to offer live sport for something like fifteen hours a day to a country not used to such a thing - it was the right game at the right time. Darts gained popularity at the same time for the same reason, but, after a few years it's popularity began to decline and it was very much second best to snooker for a long time.
Although I just don't get the attraction of modern darts myself and very rarely watch it, the game has done really well to overcome that decline it suffered and has emerged stronger in a way snooker hasn't been able to match. Others on here would know better than me, but it seems that darts has become more popular than snooker mainly because of the personalities of some of the players and the fact that the crowd seem to be having such a good time - in some ways the darts itself is incidental.
On the other hand, snooker takes itself seriously and wants to be seen as a "proper" sport in a way that darts perhaps doesn't and it suffers for this in a way that darts doesn't in this age of wall to wall live sports coverage. Back in the eighties, I'd be up till 2 in the morning watching Terry Griffiths v Cliff Thorburn like the rest of the country appeared to be, but now I just pop into televised snooker to watch a frame or two every now and again - I still prefer snooker to darts, but it's far from being a must watch at this time of year.