Both give a referral partner their own tracked lead source, so deals from them are attributed automatically. They differ only in how the partner sends you business. A sign up link gives the partner a unique link they fill out or pass along, which suits most accountants, realtors and referral partners. A Referral API is for a partner who wants to push client data to you from their own system or website form. Set up one lead source per partner under Lead sources and you can see exactly which businesses and deals each one brought in, from creation through to commission, without tracking it in a spreadsheet.
Lead sources used to be called source channels. Brokers who send you deals are set up the same way, as a partner lead source.