The Seagate would have come with Windows formatting *and* partitioning (Master Boot Record). So when I get an external drive that will be used only for Macs, I repartition it (which also reformats it) as either Apple Partition Map (required to boot PPC Macs) or GUID. APM is the best universal choice for Macs, since it can boot both PPC and Intel Macs (though you have to clone Leopard onto it, you can't install directly.) If the drive is only for data, it doesn't matter which you use.
Sure you can do a simple reformat for Mac on top of MBR partitions, but then if you try to use the drive for Time Machine I seem to remember it will want to reformat it again. TM will format the drive, including the partition, properly. But then you can't use it with a Windows computer.
If you can't copy any files to the new Seagate, perhaps it was formatted NTFS? I believe that's read-only on a Mac.