The definition of "smartphone" seems to move every year. I once read a book titled, GSM Superphones (1999). What it described was a single-band digital phone with two-way text capability. At one time the definition of a "smartphone" would be any cell phone that allowed user customization via downloaded applications. Today, many phones that fit that definition would be called "dumb phones". "Smartphones" today generally have a large screen area, camera(s), are 3G (some claim 4G) and have a computer operating system (iOS, Windows, Symbian, Android, etc) that permits the installation of a number of different types of applications. The alternate "Feature phone" has the above except no or limited ability to install new applications. "Feature phones" more or less remain the way they came out of the box. The problem with "feature phones" is that the manufacturers have been pretty bad at predicting what features customers want. So, you get a phone and a year later you look at some features and ask, "Why would anyone want that"?
The two major smartphone camps are iPhone and Android. IPhone users often seem to think that everything else is déclassé. The Android operating system was developed by Google and exploded onto the market a few years ago. Android phones now have most of the apps (at least the major ones) that are available on the iPhone. This is no reason to avoid the other alternatives, Windows Mobile, Symbian & Linux. However, your choice of "cool new aps" for those platforms will be more limited than with the other two.
It largely depends on what you want a phone for if you really need a smartphone or not. However, once you have one you may discover many uses you never thought about before you started using the phone.