What is the best time of year to visit?
What do you want to do there?
Let's start by narrowing down a season ... do you want a winter holiday (skiing, snowboarding, snowshoeing etc.) or a summer holiday (hiking, canoeing, horseback riding etc.)?
Are you willing to trade better weather for cheaper accommodations?
It's like the others have said it depends what you want to do. Winter has it's own beauty but if you are wanting to hike or see some of the higher alpine terrain (i.e. around lake moraine or lake louise) then july-mid-september is the prime time. The issue there is that it's the time everyone wants to go.
I would highly suggest not doing April-June. You can get cheaper accommodation in the spring, but as the winter snow pack is melting trails are muddy and passes, alpine terrain are inaccessible.
Go March and April. The mountains look more majestic and the popular sites are less crowded. You already know how to drive in the snow.
Is the situation same by mid June or things have started to dry up?
June is the rainiest month. Upper elevation trails are still snow packed/ slushy or muddy.
The lakes Louise and Morraine have been known to keep their ice until earky June. I think June 9 is the latest ice off date recorded.
June may be the rainiest month for the town of Banff, but there really isn't that much difference between June, July and August. And according to this chart, July is the rainiest month at Lake Louise.
Chicago, if your plans for the Rockies include higher elevation hiking (the Lake Louise tea house trails, Moraine Lake, basically anything above 5500 feet, you'll want to come no earlier than late June. The best alpine hiking conditions are July through early/mid September ... which also coincides with the peak number of visitors.