Different configurations result in different wear on bearings and different gravitational effects on heads. Warranties on external drive units (incl. enclosure) tend to be shorter than internals. Further, my standard issue PC has two quiet fans, one of which draws air directly through the drive bay.
However, the important thing is to make backups, and backups to external hard drives are the most efficient and flexible, especially when compared to labor intensive DVDs (loading, unloading, stacking, labelling, sorting). Any drive might fail at any time, running or not, so reliability and MTBF are value issues, not preventative or prophylactic.
Another reason for keeping an external drive off-line is in case of viruses or malware or trojans doing a nasty on the disk. Once you take it offline (disconnect the USB), it is up to you whether you turn it off or not, but many external drive units are powered via the USB, or have automatic shutdown.
Nothing said here convinces me to alter the way I run: internal drives on 24/7, double backups to external drives (one for offsite), external on only when needed, single backup DVD disk for only the most important data and images, duplicate backup DVD disk for offsite.