As USB 2.5" hard disks draw a significant amount of power to spin up, your laptop ports may not provide enough power to spin it up.
To check, hold the Verbatim drive while you plug it in. If you don't feel it purring (like ti would with the Desktop PC), try plugging it in another USB port. If you haven't already tried it on its own, unplug any USB devices attached (e.g. USB mouse) and plug it in. I've occasionally have this issue with my WD Passport, especially when another USB device is attached or on front PC USB ports.
If it will not work on any port, get a powered USB hub and plug the HDD into the hub. Most USB hubs typically provide more power than the laptop/PC's own ports, which will likely let you use the HDD with the laptop. I know this is more awkward, but it may be the only way to connect the drive.