Ever since the new YouTube custom URLs were introduced, YouTubers have wanted to also have the equivalent shorthand vanity URL - one without the /c/ in the middle. Now channels with a new-style custom URL automatically have the equivalent shorthand URL.
A shorthand vanity URL - one without a /user/ or /c/ - is useful to use to promote your channel. In the past, that short URL was most often based on the /user/ URL for the channel. Now that's changed. Shorthand URLs are now based on the new /c/ custom URL, not the old /user/ URL.
If your channel's old and new custom URLs are different, you should check to be sure the old shorthand URL still leads to your channel.
Update 2/26/15: shorthand vanity URLs are now based on the new /c/ custom URL, rather than the old /user/ custom URL. More details
If you have recently created a new YouTube channel, you are sure to have noticed the default channel URL includes a long string of characters that are hard to remember and tricky to type. A custom URL makes it easy to share and promote your channel.