What's the difference between 1 cup dry measure and 1 cup liquid measure?
There isn't any. But that's no excuse for not using the right type of measuring cup for whatever it is that you're measuring out.
A liquid cup and a solid cup are exactly the same size. This can be easily verified by measuring a cup of water in a liquid measuring cup and pouring it into a one-cup dry measuring cup: they take up the same amount of space.
Why have different equipment, then? The primary reason is that a solid measuring cup is designed to be filled to the top, with any excess being scraped off with a knife. This is great for things like sugar and flour, but filling a cup to the brim is too messy and impractical for liquids. Conversely, liquid measuring cups are transparent, and can be easily filled to the proper lines while leaving space at the top; however, filling something up to a line partway through a cup is hard to do evenly with solids.
Another reason is that solids pack more tightly when they're given a wider space in which to spread out. Liquid measuring cups are usually wider than solid measuring cups, which can result in them holding more granular substances by weight when filled normally. This does not mean solid cups are a different size or somehow more accurate, but since the author of the recipe probably used a solid cup for solids and a liquid cup for liquids, your best bet is to do the same.
(The ideal solution—practiced in many other countries—is to measure solids by weight, eliminating the packing factor entirely.)