There are big differences between broadband internet routers versus pure wifi.
On verizons wifi device you would get slower speeds than Time Warner, additionally you must use the wep key provided on the router supplied by verizon or in the paper work. You then if the wifi device is properly operating should have the iphone scan for the wifi/mifi device (verizon router) and once found then click connect and enter the network key.
Like I said using this type of router is not like a real broadband router. In Time Warner the signal comes to the broadband modem, Time Warner registers the modem on their network of which if it is one of the old SMC wireless broadband modems you look for the smc....and when founbd wirelessly you enter the wep key they assigned you. A hard connection via RJ-45 cable requires no network key.
In that mifi device just read the manual to see how it should be set up. Please remember the mifi may not be receiving a strong enough signal to allow you to connect your iPhones or other wireless devices.
Go back to Time Warner and you will not have those issues.