Both movies deal with the conspiratorial surveillance and profiling of its citizens by the state. Both comment on the 4th Amendment laws that should protect privacy, not lead to its invasion by illegal search and seizure and intrusion. In the growing Age of Global Terrorism, both films are relevant in depicting the US (Enemy of the State) and Great Britain (1984) as police states who set double standards in protecting its citizens from their own war on terror.
1984 shows a much more bleak, dystopic depiction of the state monitoring its citizens, whereas Enemy of the State attempts to be more realistic.
Big Brother never appears in 1984; instead, we meet the double agent O'Brien. In Enemy of the State, we know from the beginning that Thomas Reynolds (Jon Voit) is the antagonist who can summon surveillance teams at the drop of a hat. Both villains target and spy on the protagonists, but in 1984 Winston never knows why. He is not in possession of state secrets, unlike Robert Clayton Dean (Will Smith), who has a tape that exposes the villain.
1984 is a satire--an attack of an authoritarian regime. It exaggerates the powers of the state to an extreme in order to point out their cruelty and oppression. Enemy of the State is an action political thriller, more escapist drama. It does not attack the U.S. government specifically the way Radford targets the former USSR and Nazi Germany.