What is the difference between . (dot) and $ (dollar sign)?

The $ operator is for avoiding parentheses. Anything appearing after it will take precedence over anything that comes before.

For example, let's say you've got a line that reads:

putStrLn (show (1 + 1))

If you want to get rid of those parentheses, any of the following lines would also do the same thing:

putStrLn (show $ 1 + 1)
putStrLn $ show (1 + 1)
putStrLn $ show $ 1 + 1

The primary purpose of the . operator is not to avoid parentheses, but to chain functions. It lets you tie the output of whatever appears on the right to the input of whatever appears on the left. This usually also results in fewer parentheses, but works differently.

Going back to the same example:

putStrLn (show (1 + 1))

1. (1 + 1) doesn't have an input, and therefore cannot be used with the . operator.
2. show can take an Int and return a String.
3. putStrLn can take a String and return an IO ().

You can chain show to putStrLn like this:

(putStrLn . show) (1 + 1)

If that's too many parentheses for your liking, get rid of them with the $ operator:

putStrLn . show $ 1 + 1

They have different types and different definitions:

infixr 9 .
(.) :: (b -> c) -> (a -> b) -> (a -> c)
(f . g) x = f (g x)

infixr 0 $
($) :: (a -> b) -> a -> b
f $ x = f x

($) is intended to replace normal function application but at a different precedence to help avoid parentheses. (.) is for composing two functions together to make a new function.

In some cases they are interchangeable, but this is not true in general. The typical example where they are is:

f $ g $ h $ x

==>

f . g . h $ x

In other words in a chain of $s, all but the final one can be replaced by .

Also note that ($) is the identity function specialised to function types. The identity function looks like this:

id :: a -> a
id x = x

While ($) looks like this:

($) :: (a -> b) -> (a -> b)
($) = id

Note that I've intentionally added extra parentheses in the type signature.

Uses of ($) can usually be eliminated by adding parenthesis (unless the operator is used in a section). E.g.: f $ g x becomes f (g x).

Uses of (.) are often slightly harder to replace; they usually need a lambda or the introduction of an explicit function parameter. For example:

f = g . h

becomes

f x = (g . h) x

becomes

f x = g (h x)

Hope this helps!

($) allows functions to be chained together without adding parentheses to control evaluation order:

Prelude> head (tail "asdf")
's'

Prelude> head $ tail "asdf"
's'

The compose operator (.) creates a new function without specifying the arguments:

Prelude> let second x = head $ tail x
Prelude> second "asdf"
's'

Prelude> let second = head . tail
Prelude> second "asdf"
's'

The example above is arguably illustrative, but doesn't really show the convenience of using composition. Here's another analogy:

Prelude> let third x = head $ tail $ tail x
Prelude> map third ["asdf", "qwer", "1234"]
"de3"

If we only use third once, we can avoid naming it by using a lambda:

Prelude> map (\x -> head $ tail $ tail x) ["asdf", "qwer", "1234"]
"de3"

Finally, composition lets us avoid the lambda:

Prelude> map (head . tail . tail) ["asdf", "qwer", "1234"]
"de3"