In your case, the problem is the file system on the drive as opposed to how it is connected to your Mac.
The drive is most likely formatted as FAT32 system. This is a typical partitioning format that is supported by practically all computers (windows, linux and mac os). FAT32 limits file sizes to 4GB and device sizes to 2TB (or 16 TB for 4 KB sectors). For you to be able to store a file greater than 4GB in size, you'd need to reformat the drive to either ExFAT (file size is limited to 16EB, or 16 BILLION TB) or a MacOS partition format (called HFS plus, file size limited to 8EB, or 8 BILLION TB). I'd recommend the MacOS partition format for it's reliability, but only if you're working in a strictly Mac OS environment. HFS plus is NOT supported by Windows. If you need to transfer files between computers of various OSs, I'd recommend ExFAT.
Note: Changing partition formats WILL DELETE ALL DATA ON THE DRIVE, so make sure you back up all the data on the drive before you start.
- Back up all your files on the USB drive onto a secure location on your Mac.
- Open Disk Utility by searching it in spot light or opening it from /Applications/Utilities
- Select your desired drive on the left side.
- Go to the "Erase" tab.
- Choose your desired partition format from the drop down menu, and name your drive as you please.
- Click "erase"
- Restore the files back to the drive.