They are just synonymous. Duplex scanner indicates the US console that has both Grey scale (also called B-mode) Doppler capabilities. Doppler ultrasonography require B-mode images.
Duplex scanning is the use of B-mode and Doppler at the same time. Doppler ultrasound is a generic term. You can perform spectral Doppler (look at the Doppler wave forms at one point) or color Doppler (image is color coded based on some aspect of the spectral Doppler such as peak systolic velocity)
Ultrasound scanners produced by most manufacturers these days are equipped with Doppler apparatus and can therefore be used for Doppler studies as well as the traditional B-mode scan. So these equipment that are capable of being used for Doppler as well as B-mode scan are referred to as duplex scanners. They offer the advantage of seeing the structure being interrogated in Doppler study first in B-mode before Doppler function is activated. Have you seen a scanner that is solely for Doppler? I doubt if anyone has in recent years.
Older ultrasound devices could not perform B mode US and Doppler at the same time.
Nowadays almost all ultrasound machines can do both. So the name Duplex. And even more is possible in these devices. You can perform B mode US, color Doppler and spectral analysis at the same time. It has been called tríplex. Of course, it is only possible with pulsed Doppler, not continuous Doppler.
Practically there is a great difference in the image result of Gray Sale and Duplex imaging, In the image of Gray Scale Ultrasound you can only see the smoke but with the help of Doppler you can see the fire as well. I mean in Gray scale image you can only see change in echogenicity in gray scale but through Doppler you can see the blood supply, Physiologic function can also be evaluated and ultimately the lesion tumor or any abnormality else will be decided from its activity. In the era of doppler it is injustice to use Scanner without the presence of Doppler.