(although this has nothing to do with Project-GC as such and is more about geocaching in general, I'll bite)
Technically, you don't need either. If you register an account with for instance geocaching.com, you will get access to descriptions and coordinates of caches, including a map that shows you where on the map they are. If you then print out the map or have a very good memory (or have the option to bring any form of map with any form of marking on it with you), you technically have a shot at finding geocaches. It is, however, not my understanding that this is widespread practice amongst geocachers. In fact, I don't think I've ever met someone who have done this for more than a cache or two.
If you choose to get yourself a device (be it a smartphone or a dedicated GPS receiver), you will probably find yourself much more successful in finding geocaches and you will probably also find it a lot more convenient, as both app programmers and GPS receiver manufacturers have gone to great lengths to make it easier for their customers/users to geocache seamlessly.
There is a lot to be said about which device you should be getting if you choose to buy one. The main points are:
- Ease of use - does it do what you want easily, effortlessly and seamlessly ?
- Utility - can it do/will you use it for other things than just geocaching ?
- Price - considering the point above
- Battery time - also considering point 2
- Screen size - can you see what you need to see ?
- Ruggedness - will it stand up to the challenge of being brought on the trail ?
There are obvious differences between smartphones and dedicated GPS receivers. Most people I know have and use both, but for different occations. It ususally comes down to the fact that the tradeoffs on the above mentioned points are different for the two categories of devices and having a device from both categories allows you to get the best from both worlds.
I wish you good luck with your pending purchase and welcome to geocaching :)