GDPR splits the world into controllers, who decide why and how personal data is processed, and processors, who do it for them. A SaaS product processing customer end-user data is almost always a processor — which means every serious European or privacy-conscious customer will put a DPA in front of you before go-live. The agreement obligates you to process only on documented instructions, keep the data confidential and secure, help the controller answer data-subject requests, and disclose the subprocessors you rely on.
The two get conflated because both are “the privacy contract.” A BAA is HIPAA-specific: US health data, covered entities, business associates. A DPA implements GDPR: personal data of people in Europe, any industry, controller-processor mechanics, and typically an annex of technical and organizational measures plus international-transfer terms. Plenty of health-tech companies sign both — for different data, under different laws.
Your counsel owns the terms; your security program owns their truth. The measures annex you attach — encryption, access control, testing, incident response — becomes a commitment customers and auditors check against reality, which is why DPAs and GDPR programs get built together rather than sequentially.