What is the difference between firing and triggering a thyristor?

There is no difference. The terms "firing" and "triggering" both mean exactly the same.

As long as the voltages applied to both the anode and the gate have the correct values and polarities for a particular design of thyristor, "firing" or "triggering" will cause current to switch on between the thyristor's anode and cathode. The current will then continue to flow for as long as the correct source voltage polarity is maintained onto the anode.

Another opinion

If the input signal is given to a thyristor, it is said to be gated. Because of gating pulse the thyristor gets triggered ( cause & effect relationship). Once the thyristor is triggered the charged particles starts moving inside the different layers. A noise is created called "SHOT NOISE", due to the movement of charged particles. Heat energy is dissipated due to this movement of charged particles. Once the thyristor reaches this stage it is said to be fired. Hence, there is a minute difference between these three terms. But all these terms are being used interchangeably.