Could be due to a number reasons - locate the starter solenoid, which is the biggish lump of plastic with 2 very thick wires bolted to its terminals, and a single thinner wire (or maybe two) on a push-on terminal, and put your finger on it to see if you can feel a click when you push the start button - if you can feel a click, check that there is 12v on one of the terminals, which should be there all the time, ignition on or off, as it comes directly from the battery + ..... when it clicks, i.e. when the start button is pressed, there should also be 12v on the other terminal - if all this checks out, then suspect the connection to the starter motor (one of the two thick wires, the other one should go straight to the battery +) or it could be the starter motor itself ..... If you can feel no click in the starter solenoid, then put a voltmeter on the thinner, usually white/red wire that goes onto a small terminal on the starter solenoid and see if 12v appears on it when you press the starter button - if it does, suspect the solenoid itself, or its ground connection - if you don't get 12v on the white/red wire, then the problem is likely in or around the headlight relay, because the white/red wire is fed from the headlight relay, which sends 12v down it to the starter solenoid - the fact that you do hear a click means that the headlight relay is operating, but there may be a loose or dirty connection on the relay end of the white/red wire - is the headlight working ? - if so, then 12v is reaching the headlight relay, it's just not switching the 12v to the starter solenoid, either due to a bad connection as mentioned above, or rarely, a fault in the relay's internal contacts. There's one other possible cause I can think of, which is the infamous 'click of death', a feature of early EFI models caused by the ECU aborting the start procedure because it thinks there's not enough volts, but the fact that you changed the battery puts this possible cause out of the picture, assuming the battery you put in was fully charged ? .......
I hope you find the above not as confusing as it was to me trying to write it ......