This article explains the difference between an affiliate program and a referral program.
A referral program is used to incentivize your existing customers to refer their friends to sign up for your subscription. Referral programs generally compensate the referrer with products or discounts rather than cash. Typically referral programs provide sharing links for social networks and e-mail.
A typical referral program would be an arrangement in which your customers get their next month free if they refer three of their friends.
An affiliate program is a way to compensate marketers (affiliates) for generating new sign ups to your subscription. Affiliates are generally not customers of your subscription. Typically an affiliate is compensated in cash for each signup they generate.
A typical affiliate arrangement would be a blogger who receives $5 per sign-up they generate by advertising your subscription to their audience