Confused over mobile results v's desktop results......

Can anyone explain this?

I'm targeting a particular business which I would like to approach and build them a mobile website, but in doing my research on the keywords they are targeting I found something that I really need an explanation to.

The business has a normal website but DOES NOT have a mobile website, however, when I type in the main keyword they are targeting using my desktop pc, they are appearing on page 1, but if I type the same keyword into the browser in my iPhone, they DO NOT appear on page 1. In fact they do not appear in the top 6 pages, and I didn't even bother looking any further to see what page they appeared.

Whilst I was searching on my iPhone I also clicked the 'View Mobile in classic' to see if the business website appeared and it did, which proves that something is preventing the site from being displayed in the mobile search results in google.

Does anyone know why this maybe happening? I'm trying to understand why, so that if I manage to get this business as a new client and I build them a mobile site, they will get traffic to it, because right now they are not showing on the mobile search results, but they do in the desktop ('classic' view) results.