Great question!

I’ve actually been thinking about this a couple years before I started getting into ASOIAF.

Here are some things I think Renaissance fantasy should emphasize to set itself apart from High Fantasy or medieval fantasy:

• Different geographies: High Fantasy/Medieval fantasy tend to be set either in rural countrysides (following the pastoral tradition) or vast, untamed wilderness (following the Romantic), largely due to Tolkein’s anti-modernism. The Renaissance very much was an era in which cities and commerce and finance were starting to become important, rather than just subsistence agriculture. 
• Different societies: 

  High Fantasy/Medieval fantasy hasn’t traditionally interrogated class particularly well, and so you tend to get idealized images of happy peasants, rightful kings, and brave knights. (It’s only in recent years with the rise of deconstructionist fantasy that we’ve started to question this stuff.) But in the Renaissance, you start to see merchant families and guilds not just exerting political influence, but outright running city-states.

• Different politics: rather than just kings and lords, “you’ve got various forms of Republics, mercantile city-states, and petty princedoms, all of which gives much more scope for ordinary people to do important things.”
• Different cultures:  rather than an emphasis on the ancient and the eternal, there should be an emphasis on cultural change. “an explosion of knowledge, with a bubbling ferment of science, arts, literature, philosophy, history, political science, and a roster of geniuses whose human brilliance is much more appealing than the aloof [I think I was leaning towards alien or inhuman, without really putting my finger on it] other-ness of a Merlin.”
• Cosmopolitanism: in part because of urbanism and in part because of increased trade, you have a lot more cultural diversity, so in my mind Renaissance Fantasy ought to involve a melange between many different cultures beyond Expies of white Europeans, with cities full of immigrant workers, foreign merchants and diplomats, imported goods and ideas, a sense that the city is part of a global network. 

Hopefully, Renaissance fantasy should help us move beyond repetitions of the Return of the True King by way of the Hero’s Journey, and allow us to tell other kinds of stories.