同時 and 一緒: coinciding in time and doing something together
Compare simultaneous starts at two venues with traveling together, keeping the shared departure separate from arrival times.
Events that coincide in time
同時, read どうじ, identifies events occurring at the same time. 二つの会場で同時に始まりました。 says that the activities began simultaneously at two venues.
音と光が同時に出ました。 says that sound and light appeared simultaneously. Different kinds of events can be linked by time.
People or actions joined together
一緒, read いっしょ, can express doing something together. 友達と一緒に帰りました。 says that the person went back with a friend.
一緒に計画を立てましょう。 suggests making a plan together. It does not require every planned action to be performed at the same instant.
Which events share their timing
一緒に出発しましたが、到着時刻は違いました。 says that the people departed together but arrived at different times. Do not extend the departure relationship automatically to arrival.
同時に申し込んだ二人は、別々の部屋で参加しました。 says that two people who applied simultaneously participated in separate rooms. 同時 does not specify a shared location.
Were the arrivals also simultaneous in 一緒に出発しましたが、到着時刻は違いました。?
Show the answer and why
No. The people departed together, but the arrival times are explicitly stated to have differed.