As someone who's lived in both DC and Chicago, I'd say that either is a fantastic destination for 4 days.

In terms of museums, DC wins hands down...although Chicago's are great, you have many more to choose from in DC.

In terms of shopping, Chicago probably wins by a nose but these days 90% of the stores you'll see in tourist areas are chains, so there isn't a lot of variety from city to city.

If you're physically active, both Chicago and DC have a lot to offer. Chicago is flatter, which might be preferable if you like to bike and run. You can sail in both. Chicago has the lakefront, DC has the C&O canal, and each is awesome for walking, jogging, biking, etc. (To be clear, DC has the Potomac River, too, but the canal is probably a more popular waterfront destination if you're working out. Obviously there are plenty of places in both cities that aren't by the waterfront where you can get a good workout.)

I haven't eaten out in DC much recently, Chicago definitely wins if you're looking for the better-rated restaurants. That said, both offer an eclectic mix of various types of cuisine.

In September you'll find pro football and baseball in both cities.

DC's weather is likely to be warmer than Chicago's in September, though in either you could still get very hot weather, mild or even chilly temps in the early fall. In general, DC has warmer winters than Chicago, but hotter and more humid summers.

Both are very walkable cities if you're focusing on the most commonly visited tourist attractions.

Both have great zoos, though only DC has pandas! (And we'll probably know in the next day or two, but may have a panda baby in September.)