Hi Tom,
I would never throw away traffic unless it came from a "questionable" source.
If the reason for the domain switch is because the domain more accurately reflects the client's business, then the existing traffic is also likely to be relevant. This gets you "one foot in the door" in terms of conversion.
If you still have the existing site loaded, then you are in a good position, because you already know the URL and content of the existing pages.
My recommendation to the client would be to do the work to salvage the existing traffic for the domain. How well you do the job will depend on how large the existing site is, how much traffic it has and how much work the client is willing to pay for,
The process would be to add 301 redirects for all existing URL's on the new domain (that will not exist once the new content is added) in addition to 301's for all of the pages on the domain that you are transferring from. The best way to do this is to ensure that each is redirected specifically to a page containing content that is relevant. Obviously, if the old site is large and/or the budget is small, this will be more work than you are getting paid for. In that case, you have the option of redirecting groups of pages to a relevant category page, or in the easiest, but least advisable, just redirect them all to the home page.
If the existing traffic to the new domain is insignificant, then you shouldn't lose too much sleep over just redirecting them all to the home page, but if there are some visitors there, I would explain to the client that sending them to a relevant page gives him a chance of converting them.
If you choose not to redirect any of the pages you are just setting up a situation where you constantly play "catch up' on the 404's - you cannot know how long it will really take for every link in existence to be clicked by someone. Don't forget that the biggest link source will be indexed pages in search engines. These don't always get de-indexed in the time frame we expect, so there is no correct time period within which links will just go away.
I wrote a blog post on this recently after doing a "salvage' job for a client whose designer installed a new site version and killed off all his old pages without redirecting anything. While your situation is different, It talks about redirecting for relevance, so might be worth a read.
Hope this helps
Sha