They look the same, they sometimes act the same.
It's hard to tell the difference between a router and an access point, especially since one can easily act like the other from a customer perspective. But if you really understand what they do, it's easy to understand the difference.
is your link to the outside world. All the information in your network stays in your network and it all goes through a single device when it goes out to the internet. This device is part tollbooth, part traffic cop, part automated direction-giver. It knows who is making a request for information from the internet and tags that request as it goes out, so when it comes back, it can be sent to the right place. If happens millions of times a second and it's so reliable we don't give it another thought.
An access point
provides wireless access to a wired network. It's the thing that provides your Wi-Fi so connections can be made between your wireless devices and the router, or between any device that's on your network. It makes a transparent link between a wired local-area network and the wireless network it creates. It doesn't communicate with the internet directly.
...and about now you're shaking your head
because you have that one box on your desk and it provides Wi-Fi and also connects you to the internet. So which is it? Access point or router? Answer: It's both. Most consumer routers are also access points and switches (meaning they can take more than one wired connection.) In the world of the big commercial stuff, though, the devices are often separate.
It's common to put access points throughout a large building so that you can get Wi-Fi everywhere, and you wouldn't want those devices to be routers because they don't form a bridge between you and the internet. There's usually only a need for one router unless you use multiple internet service providers. Standalone Wi-Fi access points can provide wireless networking to areas without it or take a non-wireless device like an old printer and make it wireless.