The difference between a smartphone and a featurephone is that a smartphone is much closer to a full computer. Here's just a off the top of my head list of things they can do:
1. Lots and lots of games (even with excellent graphics).
2. Everything a featurephone can do (calls, texts, contacts).
3. Browse the internet with a full browser.
4. GPS navigation/maps. Most come with software that will not only tell you how to get somewhere, but will follow you as you drive and give you step-by-step directions, and even modify the directions if you miss a turn or what have you.
5. Play music, even streaming such as Pandora or Spotify.
6. Play video, even streaming such as Netflix.
7. Synced calendar/contacts/etc.
8. Some support voice commands. Android, for example, allows one to call a business or person just by speaking the name, or phone number. It will automatically look up the closest business that matches. It can also send text messages, set the alarm clock, start GPS navigation to a location/business, etc.
9. Social apps, such as Twitter, Google+, Facebook, etc.
10. Messaging apps, with AIM/MSN/Facebook/GTalk/etc., via a variety of apps (Trillian or otherwise).
11. Most have decent cameras as well, with the ability to record HD video.
Most of this is accomplished with "apps", which is similar to a program on your laptop. There's a market for the apps, and many are free. This is only a small selection of the types of things they can do. Most smartphone markets have thousands and thousands of apps.
In short, if you think that displaying the time or having a calculator is something spectacular, you need to get your hands on a smartphone (find a store and play with one). Even the cheapest featurephone will do that. Smartphones are much, much more.