"Carat" refers to the weight of a diamond, not it's size. Diamonds and moissanites differ in their density, so their weight, as measured in carats, differ.
Moissanites are less dense and lighter than diamonds
A moissanite of the same weight as a diamond will visually be larger than the diamond of the same weight.
For example, a 7mm round cut moissanite's weight is approximately 1 carat. A 7mm round cut diamond will weight approximately 1.25 carats. Note that all diamonds are different and depending on how they're cut, this comparison might change a bit.
So, while a 1-carat diamond will weigh the same as a 1-carat moissanite, the diamond will be slightly smaller in size compared to the moissanite.
To help buyers estimate a moissanite's weight relative to a diamond's weight, moissanite measurements are given in size (mm) as well as its diamond-equivalent weight, or DEW, to help buyers compare them more easily.
For example, the 7mm round cut moissanite is 1-carat or 1.25 DEW.