Is there less vertical space for an engine under the wing of a 737 Max than for an A320neo?
The short answer is:

Yes, there is less vertical space under the wing of a 737 Max compared to an A320neo.

In the image below I have put the two airplanes side to side and the height difference is very apparent. Please note that the images are from the respective predecessors of the aircraft. The only thing that has changed regarding the height is that the 737 MAX has slightly larger tires than the 737 Classics, which should raise the MAX by 2 inches.

The reason for the lower height of the 737 is that it was developed a lot earlier when engines were a lot smaller in diameter. Therefore, they designed shorter landing gear. Choosing longer landing gear would have seemed unreasonable because this comes with a significant weight penalty. Redesigning the landing gear now would basically mean designing a completely new aircraft, because it would interfere with so many other systems.


Image Source A320, Image Source 737

It is difficult to accurately compare distances in photographs, because they come with all kinds of distortions. I have resized the images so that the main gear track is according to scale. Therefore the height comparison of the wings is fairly accurate.