There are many different reasons for why the experience a user has on a mobile site is different than that of a user on a desktop site. Firstly the scale is a big part. The size of a mobile phone is much smaller than that of a desktop so already the design of the site has to accommodate the amount of space provided. This tends to make mobile sites much more simplistic and “symbol based” due to the lack of space provided.
Another different is the way the devices are used. Desktop sites tend to be controlled by a mouse where as the mobile site is controlled by user touch in the form of a touch screen. This really affects how key it is for sites to be responsive depending on the way they’re being used.
A third difference is on desktop sites when navigating it often times feel better to go between pages where as on mobile it can be quite frustrating to go from page to page on a site due to how tedious and unstable touch devices can be.
A fourth difference is more of an umbrella idea in that both versions of a site should be very easy to tell as being the same. For example facebook does a great job of this. Both the app and website are very consistent in how they display their information as well as it’s visual theme. Personally I also find that the mobile app is a much better user experience than that of the desktop which is another topic that relates with desktop vs mobile sites. Often times there will be on that is superior to the other and this just depends on the subject of the site and what it’s used for/can do.