Newegg has the better selection on PC components (sold directly by them), and their website has a much better search, layout, pictures, and specifications. You can also get free 2-day shipping with Shoprunner if you have an Amex card. Plus, they take Paypal. Newegg usually has lower prices on components (especially motherboards).
Amazon has much better customer service, and has free 2-day shipping with Prime Membership. They also offer same-day delivery on some items depending on where you live. Their website layout and search options are a pain for computer parts. They sell virtually everything on the planet, however a good portion of the selection is only offered by third-party sellers.
Newegg reviews are annoying. They filter reviews, and many are written along the style of "OMG IZ SO EPIC IT RULZ!".
Amazon reviews are usually more 'grown up', and if someone writes a review that is false or just plain stupid, you can comment on it.
That said, I usually order about 80% of my parts from Amazon.