FWIW, a comparison of the IGs for the TG1 and TG4 show these differences:
The TG4 has a connection to your RJ31X and will monitor for line fault, the TG1 does not connect to the land line at all and therefore has no fault monitoring.
The TG4 has two relays that can be connected back to your alarm panel zones to allow the panel to monitor the TG's operation and status, things like AC Fail, low battery, no service, radio failure and on a TG4, line fault. The TG1 has only a single relay. For both, one can program the TG as to which status indicators will be OR'd together to report.
The TG4 can interpret various alarm panel reporting code formats and can auto detect the format. The TG1 handles Contact ID format only.
The TG4 can be setup so communication is via the land when it's available and cellular as backup or vice versa. The TG1, with no land line connection, is only for primary connections.
In either case, the interface from the alarm panel to the TG1/4 is via the phone cord. The TG appears as a phone line to the panel.
There appears to be various price plans (hopefully someone with access to that info can provide some relative difference in the costs or MSRP) and they are dependant on the amount of traffic. So if you have your panel reporting various open/close events or arm/disarm events, that will drive up the monthly charges.
Hope this helps out... On to read up on the TG7 and anynet now.