AF1 is designed to withstand a nuclear environment and is to be used for reconstitution. It has many different forms of communications equipment and can communicate to the forces even during the trans attack phase. The communication equipment is also used in day to day operations to ensure the president can direct the Government during peacetime events.
A Gulfstream IV does not have the ability to carry that equipment nor does it have the performance to keep aloft for the extremely long periods of time required if things go south and we have a nuclear exchange.