The most important difference is that customer satisfaction focuses on meeting expectations that lie in the past. Satisfaction shows what your customers think about your delivered product or service.
Loyalty does not focus on the past but rather on the future and on the degree to which a customer stays faithful to an organization. The loyalty of your customers can be evaluated in different ways: by means of a periodical survey or with the help of continuous feedback. Loyal customers are the best indicator for a stable and healthy future because of their intention to either repeat their purchase or buy extra services or goods from you.
The Net Promoter Score (NPS) is an important KPI and management tool for measuring customer loyalty.