Here's a nice recent blog post about various issues you have to deal with to even set up a EKS cluster in AWS:
https://medium.com/@dmaas/setting-up-amazon-eks-what-you-mus...
I get the impression that compared to GKE, EKS is not yet fully baked as an easy to use product. It still involves too much manual fiddling around (e.g. see above) and operational complexity.
I'd advise any clients of mine who want to run Kubernetes in the cloud to stick to GKE for now. The lower prices on GKE are a nice bonus also.
That said, I don't discount what Amazon can do in this space. If they make the setup/admin experience as easy as spinning up a VM on AWS then Google will have a serious competitor. But not yet.
BTW, for smaller shops and solo devs, I hear good things about the upcoming Digital Ocean Kubernetes service. It's in beta right now:
https://www.digitalocean.com/products/kubernetes/