If you have items you can count, you need to use “fewer” to follow the standard usage guidelines. (What constitutes “standard” is up for much discussion, as any web search will readily confirm. I use it to mean “no one will look at you funny for using this, regardless of where you’re writing it.”) Those signs at the market that read “10 items or less” drive me batty, personally. I can count ten items. If I have fewer than 10, I’m okay. If I have eleven, I need to get in another line. (Unless the checker tells me it’s okay because there aren’t many people in the line.)
If you have an uncountable noun (like “intelligence” or “ability” or “music”), you need to use “less” to follow the standard guidelines. “John is less intelligent than Jake.” (John may have fewer IQ points. “Points” are countable. “Intelligence” is not.)
This job took less time than the last one.
This job took fewer hours than the last one.
“More” doesn’t have the same problem: You can have more time, and you can have more minutes. (English is fun, remember?)
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.”