Let's start with the definitions. Customer satisfaction is a self-reported measure of how much customers ‘likes' a company and how happy they are with goods purchased or services obtained from the company. Customer loyalty, on the other hand, is a company-calculated metric of likelihood to purchase again or not defect to a competitor.
Many of the differences are evident from the definitions. Customer loyalty is measured from the transaction data a company already owns. It's a testable number, different for every customer, and it can be updated frequently.
Customer satisfaction depends on responses to surveys. Because not all customers respond to a survey (in fact, most don't), the results are presented as a single number, which represents the satisfaction of the population as a whole. Customers don't like to be pestered with surveys, so the results are usually updated annually.
Beyond these obvious differences are more important distinctions. The first is temporal. Customer satisfaction is a backwards-looking metric, very tightly aligned with how a customer feels about his or her last transaction. It's quick to grow, and it can disappear even faster after an ugly service transaction or defective product. Customer loyalty is forward-looking, an indicator of future behavior. It's slow to grow and slow to decline, making it much more suitable to predict future behavior than customer satisfaction.