Some thoughts....
Does the Seagate drive have an external power supply?
Are you plugging that in when you connect it to the Mac?
If you are NOT plugging in the power supply along with the drive, the Mac may not be supplying enough "USB bus power" to get the drive going.
As someone else suggested, try a different USB connecting cable.
A bit of advice I offer elsewhere:
Too often I've seen posts (like yours) from people who use a "cross-formatted" drive on both a Mac and a PC, then suddenly start "losing their data" on the Mac side of things.
For a drive that is to be attached to the Mac, that you store important stuff on...
... it should be in -Macintosh format- only (HFS+, journaling enabled, GUID partition table).
If you need a cross-platform drive to share things between a Mac and a PC, use a "dedicated device" for that purpose, and ONLY for that purpose.
A USB flash drive works fine for such things.
But again, if it's IMPORTANT stuff on the Mac, keep it on a Mac-formatted drive.
My opinion only. Others will disagree....