The Peltier effect does not just involve heating, if you pass current through a loop made of two different materials one junction warms up but the other cools down. Joule heating could never explain the cooling of the other junction.
The reason is Joule heating is due to the scattering of the conduction carriers (electrons in metals) so they ultimately impart some of their energy to the vibration atoms of the conductor ie heat. In The Peltier effect the distribution of the charge carriers is altered producing the heating at one junction and cooling at the other.
There are several differences between the two effects including Joule heating can't be reversed (heating a conductor will not make current flow) but the Peltier effect is reversible (keeping the two junctions of the loop at different temperatures will cause current to flow – this is called the Seebeck effect). Also the Joule effect is much larger except for very very small currents and tends to mask the Peltier effect unless very careful experimental conditions are set up.