Almost all our clients use both.
Adwords will get a new site noticed before it has got anywhere in the rankings, while SEO gives you that long term free traffic.
There are costs to both: Adwords you pay per click, so the ROI is trackable. Any time you are not making money on that ad group, it needs to be changed so it does, or dropped (although a 'loss leader' is an option).
SEO costs are harder to track, but are primarily time in writing content, adding pages, getting links, etc. There is also the option to pay someone else to do this, and you can also pay for some helpful directory listings (Yahoo being the biggest and one of the more expensive ones)
SEO can easily take 6 months, depending on competition and the history of the site - adwords can be up and running in a day or two.
One interesting thing not mentioned: a site or page well optimised for a keyword will get a better quality score, and be cheaper in adwords for the same placement.