The words in most scenarios are otherwise indistinguishable but I'll try my best to explain the subtle differences. "Ought to" in most context refers to what "other people" think or what is "most recommended" but doesn't necessarily imply a will to do so, it is also far more formal. Whereas "should" implies an action that is preferred or demanded but may not need to be followed through, it is just lightly implied or used as a base of reference. For example: "We should leave first thing in the morning." "We ought to leave first thing in the morning." The sentence with should is a bit weaker, with the statement being a suggestion to a certain degree. This is why "should" is more commonly followed up with "probably" when in conversational tones, but no one would say "we ought to probably" because it is far more heavily implied. "Ought to" in this example that outside forces would also recommend that the group leaves in the morning, but that the speaker might not personally wish to but is aware it is recommended in general to do so. A more specific example is someone lying on a couch and thinking to himself "I should go to the store, tomorrow." compared to being scolded by their parents for not getting groceries and thinking "I ought to go to the store, tomorrow."

They have the exact same meaning. Ought to is more formal and not as commonly used as should.