Essentially, Aごとに is "for every A" or "for each A", whereas Aおきに means "at intervals of A" or "placing intervals of A".
Aごとに is relatively straightforward; 1時間ごとに飲む means drinking tea every hour, like 9:00, 10:00, 11:00 and so on.
On the other hand, 1時間おきに is ambiguous and tricky. If you think of "hour" as the smallest unit in your mental model, saying 1時間おきにお茶を飲む means that there are non-drinking hours between two drinking hours. In other words, it means drinking tea like 9:00, 11:00, 13:00. However, if you consider minutes as the unit in your mind, 1時間おき can be the same as "at intervals of 60 minutes", which would be like 9:00, 10:00, 11:00.
Similarly, when someone says 2日おき, people who think of "day" as the smallest unit would imagine a pattern like "5th, 8th, 11th", and this is the traditional and "technically-correct" interpretation. However, it is known that more and more people have started to equate it with "at intervals of 48 hours" and think of a pattern like "5th, 7th, 9th".
This is a well-known risk of using おき, and it's safer to avoid it in precise discussions. Nowadays, even 1日おき is not always understood as "every other day", so you need to be careful to avoid misunderstandings.
Note that the difference between おきに and ごとに is not about strictness. In your case, you cannot say "My next cup of tea would be anywhere between 10 am-11am for ごとに", and the same is true for おきに. How punctual you have to be depends on the situation, but if you are told "every hour" and leave a 90-minute interval, it's likely to be seen as too lazy.
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