Help with a new computer and the older external hard drive

Based on those symptoms, I'm guessing that

• The drive is formatted using NTFS – a Microsoft-proprietary filesystem that macOS can read, but cannot write without the aid of third-party filesystem drivers
• You installed a third-party NTFS filesystem driver on your old Mac, but not on your new one

You should be able to check the format of the hard drive with Get Info (in the Finder) or with Disk Utility.

If this is the case, you have two options:

• Get another external drive, format it using a Mac filesystem (HFS+ or APFS), and copy all of the files from the NTFS-formatted drive to the Mac-formatted one. (Once you're sure the files are all copied safely, you could reformat the old drive using a Mac filesystem and use it as a backup drive.)
• Get a third-party NTFS driver for your new Mac, one that is compatible with your new Mac. (If one came with your old drive, it might be too old to be compatible with the new Mac.)

Sometimes NTFS filesystem drivers for Macs come on new, NTFS-formatted "PC" hard drives. Other times they are sold separately, as with Microsoft NTFS for Mac by Paragon Software .

It's possible that some of the ones bundled with "PC" hard drives might be restricted to work only on that brand, or model, of hard drive.

There are some free drivers. Each of these vendor Web sites just "happens" to claim that its own vendors' driver is the best. Some of the free drivers are listed as having the drawback of corrupting user data – which for me is a rather major drawback in a filesystem driver!

https://iboysoft.com/ntfs-for-mac/free-ntfs-for-mac.html

https://toolbox.easeus.com/ntfs-mac-tips/free-ntfs-on-mac.html