each returns the original object. It's used to run an operation using each element of an array without collecting any of the results. For example, if you want to print a list of numbers, you might do something like this:
arr = [1, 2, 3, 4]
arr.each { |n| puts n }
Now, that puts method above actually returns nil. Some people don't know that, but it doesn't matter much anyway; there's no real reason to collect that value (if you wanted to convert arr to strings, you should be using arr.map(&:to_s) or arr.map { |n| n.to_s }.
map returns the results of the block you pass to it. It's a great way to run an operation on each element in an array and retrieve the results. If you wanted to multiple every element of an array by 2, this is the natural choice. As a bonus, you can modify the original object using map!. For example:
arr = [1, 2, 3, 4]
arr.map! { |n| n * 2}
# => [2, 4, 6, 8]
