Short answer:
The Lean Canvas is simpler and less complete than the Business Model Canvas aimed at helping startup validate their business idea according to the lean startup process.
Not so short answer:
The Lean Canvas was designed for startups and has components specifically targeted to support the lean startup process. Lean Canvas is utilized to validate your ideas and concept. Companies often move to the Business Model Canvas when they have validated the business or are already a traditional business.
Business Model Canvas tries to provide a complete business model, which can be used for testing and search in lean startups. The Lean Canvas is more focused on being a one-page summary with "simple" business models.