It depends on whether you’re talking about an uncountable thing or a countable one. It’s really that simple.
If, for example, you are talking about click-through rates for a blog post, you might say: “This post garnered a greater amount of interest than the others, based on the high number of clicks.” You’re talking about the uncountable noun interest in the first clause, and the countable noun clicks in the ending phrase.
You could just as easily say: “This post garnered a greater number of clicks than the others, showing a high amount of interest.” Here the concepts are reversed. Clicks is a countable noun concept, as we just established, so you use “number” to refer to it. Interest is still uncountable (also as just established), so you still use “amount” to refer to it.
The use of “greater” or “high” doesn’t much matter; one can have a greater number, or a high number, and be equally correct in saying so. The same goes for a greater or high amount. If you can count the thing you’re talking about, use “number.” If you can’t, then use “amount.”