A smart phone was a type of mobile phone.
Rory Williams had one in 2008. It was also a camera phone. He used it to take photos of patients which he had seen around the village of Leadworth, who were supposed to be in a coma. (TV: The Eleventh Hour)
From 2013 on, Clara Oswald had an iPhone. (TV: The Time of the Doctor, et al.) While in Clara's form, Bonnie used the phone to record video and send and receive video calls, while Clara, connected to Bonnie's mind, sent text messages to the Twelfth Doctor without Bonnie realising. The Twelfth Doctor also has an iPhone at this time. (TV: The Zygon Inversion)