A WD hard drive can be detected by one Macbook but not another
The common cause of a bus powered drive not operating is the computer isn’t getting enough power through the cable you are using to power the unit. 


USB external hard drive is mounted on one MacBook Pro but not another
External hard drive not recognised by MacBook Pro.
USB Hard drive has very small chance of being recognized by Macbook Pro each time it is connected, but works with other machines


Swapping cables or using a powered hub or getting a new enclosure can help triage and isolate why this happens in many cases. Also, looking for a long running fsckprocess I’m case the “good” mac is actually triggering an “unclean mount” type check. That often delays a device from mounting, but you should see it in diskutil immediately which doesn’t match your case here.