遅れる and 遅らせる: being delayed and causing a delay
Compare a delayed start with postponing departure, then read a forgotten reply without assuming every delay was intentional.
An event becomes late
遅れる, read おくれる, means being late relative to a reference. 開始が予定より十分間遅れました。 says that the start was ten minutes later than planned.
返事が遅れました。 says that the reply was late. The sentence alone supplies neither the specific cause nor anyone’s intention.
Causing the timing of an object to become later
遅らせる, read おくらせる, means delaying something. 出発を翌日に遅らせました。 says that departure was postponed to the next day.
開始が遅れる presents a change in the start itself; 開始を遅らせる presents an action causing a later start. Read が or を together with the following verb.
Causing a delay does not always mean intending it
返事を書くのを忘れて、返信を遅らせてしまいました。 says that forgetting to write the reply caused it to be delayed. Forgetting is the stated cause.
遅らせる alone does not establish a deliberate wish to postpone. Read any stated cause or purpose, such as 確認のため or 忘れて.
Does 返事を書くのを忘れて、返信を遅らせてしまいました。 report an intention to delay the reply deliberately?
Show the answer and why
No such intention is reported. The person forgot to write the reply and caused a delay; forgetting is the stated cause.