Wiktionary can often help with your sort of questions. By the way, I didn't know that this word has different British and American spellings, so I got to learn something today, too.

cosy ( , )

1. Affording and ; snug; social quotations ▼Synonym: Hyponym:
2. and I feel very cosy here in my bed.

Compare that to:

comfy ( , )

1. () . quotations ▼The robe and slippers were so warm and comfy she just fell asleep in her chair.

However, I wouldn't call my slippers cozy. Cozy is more for a warm, safe, and comfortable environment that surrounds you. When I lie on my bed it is comfy. When I snuggle under the covers it's cozy. Edit: I could easily say it's comfy under the covers, too, but I don't think that I would say it's cozy laying on top of the bed.

Loosely they are interchangeable but what @andrewdaniele said for more precise usage.

I would use cosy to describe an atmosphere (such as the atmosphere of an apartment), and comfy to describe an object (such as a couch).