Internal combustion, 250cc fuel-injected motorcycle.
Original battery is 12V 6Ah (10 hr) CCA: 130A
The new battery is 12V 4Ah (10 hr) CCA: 70A
Should I worry for anything apart from the actual capacity (1/3 less time) of the battery?
Power = volts X amps. A smaller capacity battery will generally deliver less cold cranking amps (CCA) than a larger capacity battery. By reducing the amps delivered by the battery you are reducing the power it can deliver to crank your starter, so you could have problems starting up or running under a high electrical load.
If the 4Ah battery has the same CCA as the 6Ah then you might get away with it, however you'll have less overall capacity, your 4Ah battery will not be able to crank for as long as a 6Ah. There's a reason it came with a 6Ah battery, and you should replace it with a 6Ah one, not less.
Edit after CCAs were added into the question: The 4Ah battery has almost half the CCAs as the 6Ah battery, if your bike has a starter the 4Ah battery may have trouble cranking it as it has far less power.