The assessed value isn't always lower.
As Patrick (and others) point out, assessments are broad averages, based on historical sales. When prices are rising, then looking at last year's sold prices means that the assessments generally will be lower than today's typical listing price.
On the other hand, when prices are falling, then looking at last year's sold prices means that the assessments generally will be higher than today's typical listing price. Where I live, assessments are done annually. So the 2010 assessments are based on 2009 activity. So the data are already a year out of date. There are plenty of jurisdictions where assessments are only done once every 2 or 3 years. So those are even less reliable.
Further, as almost everyone here points out, never, ever, base your offer on the tax assessment. The tax assessment is, at best, a broad average. It's not specific to any one home. The tax assessor didn't come out a few months ago and inspect the home. And where I live--and I suspect similar policies exist in many other areas--the county considers an assessment "accurate" if it's within nearly 10% of the actual value of a property. So, the county where I live would consider an assessment of $400,000 "accurate" if the true value of the property were between about $365,000 and $435,000. That's a mighty big range. And I can tell you that not all assessments are "accurate" even within those guidelines. So, you definitely wouldn't want to make an offer of $400,000 on a home assessed at $400,000 when its true value is only $365,000.
So: Have your Realtor run a CMA on the property you're interested in. Pay no more than the CMA suggests. Probably offer less. Your Realtor can provide guidance on strategies.
Hope that helps.