an access point is just as it says...an access point nothing more nothing less.Usually has a wireless component, and a wired cat5 port to connect to your network.
while a router usually incorporates a switch , LAN ports.There is also a WAN connection. There usually is a firewall, port forwarding, mac tables, DMZ, wireless, DHCP plus others. The most important part of a router is NAT. NAT is network address translation, this allows your router to take the single ip address from yor ISP and use to connect a mutitude of work stations. With out NAT , if using a hub for instance that does not have NAT, The first work station on would aquire that one ip address from your isp and any other work stations would not be able to connect. So NAT Takes that one ip and keeps track of what w/s make requests on the internet , when the requests come back NAT makes sure the request goes to the correct w/s. THEN thre is also SPI or stately paket inspection...SPI inspects each data packet that comes into your router to make sure it was actually requested by someone on the LAN and was not foged by a hacker.