Running involves considerably more impact, and every time your foot
With the elliptical trainer, your body weight is constantly supported. Youre always attached to the machine, and your feet are traveling in a prescribed arc. The elliptical trainer is not exerting the force against your body that the ground or treadmill exerts as you land with each foot strike. Your heart rate might be elevated, and you might be breathing hard on the elliptical machine, but your muscles, ligaments, tendons, and bones arent undergoing the same stresses.
The bigger question, however, is whether running for ten minutes on a treadmill and using an elliptical trainer for 45 are equal in their impact on your fitness. To improve cardiovascular fitness, the longer you can sustain exercise, the better. If that means using a lower-impact machine so you can go nonstop for 45 minutes, thats better than exercising some other way for just ten minutes. The key elements in cardiovascular fitness are time and consistency.
If your goal is to finish a local charity run, a 10k, or a marathon, you have to start running for real. Right now if you can sustain ten minutes, and youre struggling by the time youre done, your body sees that unpleasant experience as a good reason to adapt, so the same effort isnt as difficult the next time. The elliptical machine might stress the cardiovascular system, and cause it to adapt, but in order to develop the muscles and technique necessary to be a runner, you need to run.