Are there functions with a Fourier transform but no Laplace transform?

Question

Let

$$A = \text{the collection of functions with a Fourier transform}$$

and

$$B = \text{the collection of functions with a Laplace transform.}$$

What is the relationship between $A$ and $B$? Based on this Wikipedia page, any $f \in A$ has a Laplace transform that converges along the imaginary axis, so $A \subset B$. If this is correct, I don't understand why we learn about the Laplace transform separate from the Fourier transform. Is there a sense in which $A \nsubseteq B$? To put it another way, would it ever make sense to say "The Laplace transform does not exist, so we must take the Fourier Transform" for any common definitions of the Laplace and Fourier transforms?

I can suggest two possibilities that may lead to $A \nsubseteq B$:

1. Higher dimensional generalizations of the transforms may lead to $A \nsubseteq B$
2. Consideration of non-functions (distributions) may lead to $A \nsubseteq B$