I've owned both
The one I STILL own is leather, for all the reasons other people have mentioned: It's softer, breather better, wears more elegantly, etc. It's been my primary office chair for nearly 15 years, and it looks barely broken in.
One benefit of leather that I haven't seen mentioned is temperature. In a cool room, the leather will feel much warmer against your skin than vinyl. In warm temperatures, vinyl will stick to you like a band aid. Leather is just a nicer material to sit on.
One additional point. Despite the Eames' original intent, the chair never really was suitable for outdoor use. Touting the vinyl version of the chair for its ability to survive the elements is specious.
Thats true
I posted my eames chair find a couple of days ago. They are upholstered in naugahyde (essentially vinyl) and were left outside for quite awhile by the previous owners. The aluminum needs polishing but amazingly the vinyl cleaned up nicely. The vinyl seems quite comfortable. Buy vintage if you can, save money, have the original four star base, etc. That is if you care too.
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