They are the same, but we usually say 'degrees Celsius' or 'degrees centigrade', for example, 'today it is 14°C' is read out loud as 'today it [i.e. the temperature outside] is fourteen degrees Celsius'.

We would say "degree Celsius" when talking about where something (like air or water) is on the temperature scale. "It is thirty degrees Celsius." A scientist would say "a Celsius degree" when she needs to distinguish it from "a Fahrenheit degree" because they are different. She is not talking about where on the scale something is. She is talking about the amount from one (any one) to the next, on the scale. In many dialects of English, when talking about numbers, we use the plural, in this case "degrees", when talking about multiple (20 = "twenty degrees" ), decimals (1.5 = "one point five degrees" ; 0.5 = "zero point five degrees" ), zero ("It's zero degrees outside!"), but use the singular, in this case "degree", when talking about exactly one ("It is one degree outside." ). So, on the world weather report, you will hear, "It is one degree Celsius."