I see both terms used, and I was wondering what the difference is between a wifi router, and a wifi access point. Does the latter only create a LAN, with no way to connect the wifi box with a second network?
For the most part the two are used interchangeably, however, technically there is a difference and for the most part consumers only use WiFi Routers.
A WiFi router is a WiFi access point with a built in router. The router allows you to connect multiple computers to the network using a single IP address, typically provided by your ISP.
A WiFi access point is typically used only in enterprise networks where they have a larger router which routes their entire network and the access point only acts as a gateway between the wired and wireless networks.
The access point acts as an ethernet switch, has one Ethernet port and allows WiFi stations to access the same LAN the access point is in.
The router will typically have 4 ethernet ports, 1 special ethernet port called "internet" and will allow creating a new subnet for the Wifi, with DHCP, acting as the default gateway for this subnet
I would say that Access Point (AP) doesn't give you IP addresses. If you plug in to its WAN port, the IP address should be the same from it's 'upper' source (ISP, company's LAN, and hotel etc.). On the other hand, Router would provide you different IP segments. The IPs you've got, to your iPhone, iPad, or wireless devices, should be in different segment from the 'upper' source.