You will be well served by either. A few things I've noticed:
The Henery standard .22's have plastic barrel bands. It means nothing and you probably will never break them, but it just seems wrong. The Henry is built in the USA with no outsourcing. That's becoming more important to me. The Henry usually seems to have nice wood, and both the butt stock and fore end are matched at the factory.
The Browning is a very solid little rifle. I never muched liked the trigger group seperating and riding with the lever. In theory you can fire almost as fast as a semi, but that's not what a lever action is about. I find the rifle to be smaller framed for my liking. It just feels tiny.
Have you considered Marlins excellent model 39? I think it's still the same price range as a Browning, but it is an awesome rifle. It has genunie weight behind it and feels very solid. I have put alot of rounds through the Marlin and it is deadly accurate. Plus it feels like a full sized, real hunting rifle. It has a beautiful, cut checkerd stock and for end of walnut and is a really nice rifle. Just somthing to think about.