What is the difference between machinist tool chest vs normal tool chest other than the machinist one having more drawers and smaller size drawers? What would typically be stored in smaller size drawers?
The two you are comparing are different quality from different brands with associated (vastly) different price points. If you operate the drawers on each you will feel a difference, and the overall quality of construction is quite different.
Aside from drawers sized to the typical precision measuring tools you don't want knocking about with larger tools or piled on top of each other, there is typically a drawer (top center) in a machinists box specifically sized for a particular book - Machinery's Handbook - the "bible" of the trade. There is usually a security panel that covers all the drawers when the box is locked. Likewise, felt lining is standard in the drawers so the tools are not abused by the drawers themselves.
If you are gift shopping and can't grasp the more than $300 difference, contribute part of the money towards the expensive one if that was what was requested. The cheap one is not at all the same thing, if the recipient is a machinist in training (normally the only folks that need one.) The cost of the tools that go into one dwarf the cost of the box rather quickly. The job of the box is to keep those tools safe.