Record types let you offer different business processes, picklist values, and page layouts to different users. For example: You might create record types to differentiate your regular sales deals from your professional services engagements, offering different picklist values for each. Or you might display different page layouts for your customer support cases versus your billing cases.
At the other hand, Page layouts represent how an object is shown from a detail level. Lightning App Builder record pages almost always contain that detail, but also provide other features, like Flow embedding, or Chatter, or any other format.
Page layouts are close to the object, Lightning pages are close to the app.