See also this page where you can compare the different emoji sets side by side. I would say the Apple set looks nicer than twitter:

Hmm there is also a curated set of emoji as part of a ruby gem created by Github. The images seem to be web optimized.. or something? Mostly they are the apple emoji. I didn't look into it too much but looks promising:
posted by kablamo Staff3 years and 9 months ago Link

I prefer Google's over anything. The rest look outdated....
posted by <hidden> • 3 years and 9 months ago Link