Requirement:

A requirement is a capability that a product must possess to ultimately satisfy a customer need or objective. A requirement tends to be more granular, and is written with the implementation in mind.

Feature:

A feature is a set of logically related requirements that allows the user to satisfy an objective. A feature tends to be a higher-level objective than a requirement.

For example, when designing an Online Bookstore:

• Feature:
  • Online Shopping Cart
• Requirements:
  • User shall be able to add books to online shopping cart
  • User shall be able to remove books from online shopping cart
  • User shall be able to view books in the online shopping cart at any time
  • ...
  • User shall be able to start his checkout from the shopping cart

Our blog post here addresses this question as well.