I understand that both can refer to time or a period of time, but I don't really get the difference. I've been told they aren't interchangeable, and that 時期 has a relation to seasons, but unless 時期 is used exclusively when talking about the time between seasons, I don't really get it.
Yes, 時期 is closer to "season", and it typically refers to time in the order of weeks or months.
- クリスマスの時期: Christmas season (roughly November to December)
- クリスマスの時間: certain hours on the day of Christmas, e.g., クリスマスの時間を家族と過ごす
- 卒業式の時期: roughly March in Japan
- 卒業式の時間: time during a certain graduation ceremony
時期 also means "good time (in the future) for doing something", and in this case it can even refer to a time point years later.
I'll tell you the answer when the time comes.
(It may be 2 weeks or 2 years from now.)
I'll tell you the answer when time is up.
(It may be 20 minutes or 2 hours from now.)