用事 and 約束: things to do and commitments to others
Compare an errand or other business with a commitment made to someone, recognizing that being unavailable does not specify the commitment.
Having something to attend to
午後は用事があります。 says that the speaker has something to attend to in the afternoon. 用事, read ようじ, broadly covers tasks or personal business, not only arrangements with others.
銀行で用事を済ませてから、帰りました。 says that someone finished their business at the bank and then went home. 用事を済ませる describes completing what needed attention.
An arrangement with someone
友達と三時に会う約束をしました。 says that someone arranged to meet a friend at three. 約束, read やくそく, concerns an agreed action or commitment.
明日返すと約束しました。 says that someone promised to return it tomorrow. 約束 applies to a future commitment, not only a scheduled meeting.
A reason for declining and its details
今日は用事があるので、参加できません。 says that the speaker cannot attend because they have other business today. It leaves that business unspecified and gives no basis to assume the reason is false.
約束を守る means keeping a promise, while 用事を済ませる means finishing one's business. The verb–noun combinations distinguish task completion from responsibility to another person.
Does 用事があります establish an arrangement to meet a friend?
Show the answer and why
No. It says that there is something to attend to without specifying what. A meeting arrangement can be made explicit with 会う約束.