The Insurrection Act is not the same as classic martial law though.
Contrary to popular belief, what the Insurrection Act does is that while it does grant the president more military powers this still don't grant the president full governmental authority. Local law enforcement, the courts, state legislatures, and Congress would still operate like normal under the Insurrection Act. The Insurrection Act simply assist law enforcement in suppressing any riots or insurrections that gets way too out of control (like the 1992 Los Angeles city riots).
Martial law is a different story. What happens under martial law is that the civilian government is completely suspended (this includes the courts and the legislatures including Congress being suspended) and the military takes on the role of civilian government. Under this scenario, elected and appointed officials would be forced to step down from their posts and the military takes over these civilian roles (law enforcement, the courts, state and federal legislature, post offices, airports, museums, social services, etc would be under full military control) and the Insurrection Act simply doesn't grant the president this kind of authority to impose martial law.
I'm aware that Trump wants to invoke the Insurrection Act because he thinks this will grant him martial law powers (it doesn't) probably for the same reasons he probably thinks Cruise Control in a car is "Autopilot". The Insurrection Act would give Trump more broad military powers but this still isn't the same as martial law where power is rested on one central military authority under martial law rule.