Summer is when Chicago is at its best but Spring can be a great time to visit.
The main upside of Spring is the lack of crowds. Okay there will still be a lot of tourists but nowhere near the hoards that descend of Chicago in the Summer. This means less expensive hotels, shorter museum lines and a better chance to snag a table at a hot restaurant. The lines at the Shedd Aquarium can be over 2 hours long in the Summer but are usually less than an hour in Spring.
The water taxis and lake and river tours have started back up for the season. there is no better way to see the city than on the water. In peak season the boats are packed and you may not even get on one. Now they are half empty and you can just walk up and get a ticket.
The weather can be iffy. The cold waters of Lake Michigan can turn downtown Chicago into an icebox in a matter of minutes on an other wise nice day. The temperature can drop from block to block as you approach the lake.