Both phrases are used in describing the quantity of something.

Let’s get straight to the point. Use “amount of” followed by an uncountable noun, and “number of” with a countable noun.

Examples:

(a) amount of

• money
• water
• time
• interest
• information

1. We have to acknowledge that the amount of false information on the Internet has caused lots of problems.
2. The amount of time spent on playing should be spent on studying instead.

(b) number of

• people
• times
• words
• issues
• episodes

1. I’ve asked a number of people about it, but no one seems to know what had occurred.
2. They’ve been to Malaysia a number of times, so they’re familiar with some places on interest.

Note:
When you use “number of“, the nouns are always in the plural form.