I just replaced my spark plugs, wires and rotor and distributor cap, but I made a mistake and installed the wire with a different firing order. What problems this may cause to my engine?

 

Welcome to the site!
Depending on how different the current firing order is relative to the spec'd order - you will likely experience varying degrees of missing, on up through to a full-on rock-n-roll, rompin', stompin, snortin', buckin' and backfirin' event - there is no beneficial outcome leaving the order out of sorts.
Damage to the engine is a mild possibility, suggest to correct the order before trying to light off.