App vs mobile site in data use?

It depends. An app that sends a few words to its server uses a lot less data than looking at a page full of graphics. An app that uploads tons of data to its site uses a lot more data than looking at a site with a few paragraphs of text. It's like asking whether a glass container or a plastic container holds more water. It depends on the sizes of the containers. In your case it depends on the amount of data. (Looking at a mobile site is using an app, BTW.)
 
OK... As a example ... Google now has weather cards how much data would that use compared to AccuWeather? According to my AccuWeather 13.95 data and 12.95 cache...