Can I use this replacement motor without modifying the circuit?
Regarding voltage, I think there should be no problem as the datasheet for your replacement motor specifies a "suggested voltage 4.5V DC".

However, be careful that there is another important value in motor datasheets: the stall current.

Simply stated, this is the current consumed by the motor when it tries to rotate but is unable to, due to its too low torque in comparison with the effort needed for rotation, e.g. if it tries to rotate the wheel of a robot that is too heavy.

The original motor has a stall current of 0.8A according to its datasheet (you can get it from the link you provided).

Your replacement motor has 1A according to the link you provided. 

These 200mA can make a big difference for the transistor used in your circuit, a P2N2222A.

Indeed, that transistor is specified to switch a maximum of 600mA current at the collector.

So in the best case, the motor will run correctly without load, but if you provide some load that makes it stall, then the transistor will raise temperature (you could sense that by approaching your finger) and ultimately would get toasted.

Please also note that some poorly designed motors may consume a current near the stall current, due to internal friction (I have already almost grilled a transistor this way...)

So my answer is: 


you can do it for a simple experiment with no load on the motor
if you really want to do something with the motor, take another transistor with a higher Ic current