Mobile SEO, what to focus on and how is it different?

I have a site which ranks really well for certain keywords, sometimes taking up all 10 of the first results on Google with its website, social pages and related domains. However, when I do the same search on any modern phone I get results that are totally different with competitors taking 4th or 5th spots.

What should I focus on to take these last spots?

Any website that I have is already mobile friendly and passes Google checks for optimization and mobile.

Can I do anything with Facebook pages, Google+, Twitter etc. to rank these higher?

Any social networks that are really good to focus on?

Using social networks certainly helps. Having a Google+ business page and Facebook page well optimized for those keywords can get you a boost. You know, the usual: sneak in keywords in About sections, pick the right page category, use keywords in posts you make. For mobile searches, it's good to go local, since most people searching on their phone are interested in products and services that can be found close to them, "on the run". If you can, if your site is like that, add local indicators like address, map, working hours and similar.