BIP39 specifies a standardized mnemonic code for displaying or inputting entropy.

Advantages:

• anecdotally, data entry from mnemonic codes is much easier than base58, hex, or digits. This is true for many reasons: eye tracking of the current position, subvocalization, locally detecting typos.
• the redundancy can prevent information loss or simple transcription errors like transposition. (eg: handwritten digits are more likely to be smudged or lost in creases, etc…
• if we declare a standard soon in the ecosystem's lifetime, it provides better interoperability across software.
• if we use similar encoding standards as Bitcoin we can leverage Bitcoin key management tools, eg https://iancoleman.github.io/bip39/

Disadvantages:

• the word-lists are language specific.
• users may be tempted to try to memorize word lists more than 'opaque encodings' and then forget them and lose funds.
• users may be tempted to pick their own words from the list, thus having low entropy, thus having their funds stolen.

(Nathan just wrote the last two disadvantages to be even-handed, but is skeptical of both claims.)

Note: This is an attempt to break out multiple overlapping features from zcash/zcash#18.