With your test showing the old one works with jumper cables you have the partial solution-there must be a voltage drop somewhere in power lines to solenoid, for example in ignition key or, if you have automatic transmission, in the "switch" preventing you from starting when stick in drive. In a previous 740 of mine switch was located in base of stick, found when removing console. Problem was there, due to bad contact. Only occurred during real hot days.
As far as I remember the 240 may use grounding through key, whereas the 740 used positive voltage through key etc to activate solenoid-you will have to consider this too. There is in some models of at least the 740 a service lead/post coming out of left side wiring bundle on top of wheelhouse (I needed to dig somewhat in tape on lower aspect of cable to find it) which can be used for starting from engine room during servicing/trouble shooting. Look for this possibility also in your model´s wiring, because if you can start through that point it tells you whether failure is in/near key or in/near stick switch (if auto trans). Here I may be wrong but I think the similar connector in the 240 may be found near ignition coil. Maybe someone here on board can tell you more about your specific model regarding whether a service starter connector is present. Also post whether you have auto trans. To further dissect, find wiring diagram and try bypassing various points of connection/switches (I think the Hayne´s manual has some suggested points to test at but you may figure out additional points in the diagram if no immediate success through above). --
Lars