Clay is porous and loses moisture faster than plastic. This means the evaporation helps cool the soil. Having a higher rate of moisture loss means that the pot will require more water. This need for extra water gives the gardener a chance to drive excess salts through the pot and out the bottom - not something that happens as frequently with plastic pots and not at all with bottom watering systems. The buildup of salts is one of those reasons that houseplants just up and die every now and then.
Clay pots look better than plastic pots.
Clay pots are heavier than plastic pots. This is an advantage if your pots tend to turn upside down with the weight of the plant, but a disadvantage if you have to carry them around a lot. Clay pots do crack or break easier than plastic pots and they must be treated properly during the winter months or the gardener will regret their indifference. The saddest thing I ever saw was the day we smashed a 24 inch clay pot wide apart while moving it to the garden. My summer was not the same without my favourite huge pot.