Actually the Omnia is a smartphone (notice the lower case "s"). It is also known as a PDA Phone. The difference is in the capability of the phone. Non smartphones are generally known as feature phones.
Smartphones (notice the capital "S") are non touch screen phones that are smartphones.
In Microsoft parlance the Smartphone is known as Windows Mobile Standard. The smartphone version is Windows Mobile Professional. If you have a PDA (a non phone device) it is Windows Mobile Classic.
Clear as mud know huh.
This might help also.
http://en.wikipedia.org/wiki/Windows_Mobile