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An external hard drive is device which contains a hard drive inside its own custom enclosure. The hard drive inside is not meant to be removed or swapped, and is meant to be used as a plug and play device.
In contrast, an external hard drive enclosure is simply the enclosure part of this package, and you have the ability to plug any compatible internal hard drive into the enclosure. For example, you could take the internal hard drive out of a dying computer, insert it into the external hard drive enclosure, and easily access the dying computer's hard drive as an external drive.
Whether you get an external hard drive or an external hard drive enclosure, both provide you with many benefits. Any form of external hard drive is ideal for backups, since they are not on the same power supply and in the same machine as the rest of your computer, they'll be more resilient in case of a hardware failure than an internal drive. They also make it easy to share files between multiple computers - you could even use one external drive to backup multiple computers' files.
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External Hard Drive Advantages
Portability - External hard drives are generally much more portable than enclosures. External hard drives are meant for travel and to be the entire package, but lots of external hard drive enclosures are not meant for travel and don't shield the drive inside of them. If you just need a device to throw in your laptop bag to take backups, external hard drives are probably your best bet.
Price - External hard drives, as a complete package, are generally cheaper than an equivalent external hard drive enclosure plus the separate hard drive needed for it to work. Of course, if you already have an old supported hard drive laying around, it may be cheaper to use that and an external hard drive enclosure.
Size - An external hard drive enclosure that also supports 3.5" desktop size hard drives may be noticeably larger than an external hard drive. While an external hard drive enclosure which only supports 2.5" laptop size hard drives may be more competitive in size, you lose the advantage of being able to plug desktop hard drives into it.
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Ability to Access Internal Hard Drives - You can plug any supported internal hard drive into a external hard drive enclosure. If you have a dying computer, you can easily take its hard drive out, insert it into the enclosure, and back it up. If you deal with many dying computers, external hard drive enclosures are much more convenient than constantly plugging new internal hard drives into your system.
More Supported Data Connections - Many external hard drives only support USB 2.0, but external hard drive enclosures tend to have more supported options such as eSATA. These options can provide faster data transfer rates.
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Image Source: Wikimedia Commons/Datawalk, http://commons.wikimedia.org/wiki/File:Datawalker_XS25_black.jpg