An access point does not do what your router does to share the Intenet connection.

It wasn't long ago you would buy a router then add the AP. Today, they are all in one.

As to range, they should be similar or same. But an AP could let you run Ethernet to the AP which is closer to where you need coverage.

Finally, if you buy a router, you can configure it to be an AP. But and AP can't be configured to be a router.