A primary advantage of using only numbers is that they can be entered much more efficiently using 10-key.
The length of that number should be as short as possible while still encompassing the entire entity space you expect to catalog with room to spare. This can be tricky and should be given a bit of thought. A little set theory can give you the number of unique keys you will have access to, given a group of elements.
It is natural when speaking, to break numbers up into sets of two to four digits. By inserting dashes in some pattern, you can "force" the customer to repeat them in a more efficient and unambiguous manner.
For instance, 323-23-5344, which, of course, is social security number format, helps to inform the speaker where to pause when vocalizing the number. It also provides a visual delineation when writing the number and makes it easy to compare when copying the number.
I second the recommendation that the ordering system masks the input correctly so that no dashes need to be entered at any time. This should be carried through to printed forms to provide a clear expectation of what should be entered. For instance, a printed box for each digit separated by printed dashes.
I disagree that too much information should be embedded in this number especially if those attributes might change. For instance, say we give "323" the meaning of "is a nice customer" but then they call in four times with an attitude. Are we then going to change their customer key to "324", "is a jerk"? What if they are in region 04 and move their company to region 05?
If that happens, your options will be to update that primary key throughout the database or live with the ambiguity that the information embedded in that key is no longer reliable, thus rendering all of the information embedded in the keys of questionable utility.
It is better to store attributes that may change as separate fields in the database and have the customer number be a unique, unchanging key for that customer.