An extender is a very simple device. All it does is connect to a remote wireless AP and repeat the signal. It’s the functional equivalent of an amplifier. It typically doesn’t provide any routing features such as NAT, firewalls, etc., or DHCP server.
A router connects two networks together. It provides a means for users/devices of one network to navigate to the other network. Usually that’s unidirectional due to the router’s firewall. And by using NAT, it’s able to share a single IP (assigned to its WAN port) among many users/devices on the local network behind the WAN. Most manufacturers add wireless as a convenience.
So these are different devices intended to solve completely different problems. Of course, like many network devices, you can often combine them in various ways to solve new problems. For example, if the extender has a wired ethernet port, you can probably use the extender to create a wireless bridge between the router’s WAN and a remote wireless AP. IOW, create a WISP router.
As it happens, there aren’t many commercial WISP routers available. IIRC, ZyXEL has a few.
Personally I recommend a divide and conquer strategy. Find a wireless router you like, with all the features you want, *without* consideration for WISP. Otherwise, the number of available WISP routers is so small you’ll likely end up sacrificing too many other features to gain that one feature. Solve the connection between the wireless router and the remote wireless AP *separately*. All you need is a wireless adapter/bridge to solve that problem. But you might need a more powerful, long range, and probably more expensive bridging solution. Ideally you’d like the extender you already have to serve that purpose. But if it doesn’t, then you need to up the ante.
One way to up the ante would be to consider a higher grade wireless bridge, perhaps something from Ubiquiti. In fact, if the problem is bad enough, they have solutions that place adapters and antennas on BOTH properties, and are capable of transmitting over long distances using non-wifi channels.
http://www.youtube.com/watch?v=T9JKSVx0f7E