What is the difference between EGT and TIT?
EGT is just that - the temperature of the gas leaving the gas core at the jet pipe.  TIT is the temperature of the gas in the gas generator as it exits the combustors and enters the first high pressure turbine stage.  EGT will be significantly cooler - sometimes by as much as 1500°F! - than the TIT, as the enthalpy or internal and kinetic energy of the working gas is converted into mechanical work by the turbine, reducing the pressure and temperature of the gas flow as it exits each stage of the turbine.  See the Brayton Cycle model for these kinds of heat engines.

Usually the power output for a turboprop or turboshaft is measured as the torque on the power shaft turning either the propeller or rotor transmission.  Thrust on jet engines is usually gauged by the flight crew as N1, or the ratio of the current speed of the fan/LPC/LPT spool to its maximum speed.  This is a bit misleading as the actual thrust the engine generates does not vary linearly with fan speed, but that is the benchmark.

EGT and TIT are not monitored for power output reasons so much as making sure the engine does not exceed its thermodynamic limits.  Too hot a temperature in parts of the gas core will cause serious damage to the engine, requiring major overhaul and/or replacement of the entire engine or subsections.  These temperature limits can exercise limitations on power or thrust settings for the gas turbine during operation to prevent overheating.  On a free turbine turboprop, for instance, your power settings are bounded by both the torque limits of the propeller gearbox and another kind of hot section temperature called the Inter Turbine Temperature (ITT) which is the gas temperature between the high pressure gas generator turbine and the low pressure power turbine.  These limitations cannot be exceeded in order to prevent engine damage and may restrict maximum power settings at both low and high altitudes.