My usual disclaimer: I'm still quite new to pixel art, and this is only what I can quickly demonstrate in my lunch hour!
In this first image I've tried to break up some of the obvious color banding by breaking up any straight-ish lines of color. I've added in a few more leaves and tried to give those bands a wiggly, leafy edge instead. You can go much further than this. The aim is to make it look more "bushy".

This next one is quite subtle, but really gives things more of a 3D effect. Pick a leaf you want to appear 'above' other leaves nearby, then take the green color two shades down and underline it from below. You may want to underline the whole leaf, or only some of the pixels (to give the impression of it lifting out from the surrounding leaves). The overall effect is of making the leaf "pop" out of the picture a little.

I would go a lot further with this. For example, taking some of those areas of block color and breaking them up with darker shades describing more leaf shapes. This is more important with medium color values, I think. In very dark areas and very bright areas you can get away more with areas of solid color. In the former case you're emulating the low detail you see in shadow, while in the latter you're emulating the low detail you get by being dazzled by the light I guess.
You can go as far as shading individual leaves if you have the time and the inclination. At this point you want to start telling a story with each leaf. For example, shading it darker where it's coming out of the surrounding leaves, brighter where it's angled towards the sun, and then taking notice of any shadow it may be casting over another leaf. However, I don't think your style needs things to be quite so illustrative.
Edited to add: although I haven't shown this in the edit, you can "double underline" a leaf to make it pop out even more. At this point you may want to give a little thought to the shadows that leaf may be casting though.