Confusingly there is also a free SQL Server Compact Edition which is even lighter weight and meant for mobiles etc. You can see a comparison of the Compact versus the Express editions (PDF). You are free to redistribute both Compact and Express editions. If you have any of these - SQL Server 2000 MSDE (SP4) or SQL Server 2005 Express/Express Advanced you can upgrade to SQL Server 2008 Express. There is also a low cost SQL Server 2008 developer edition which provides a single license ideal for developing your own SQL Applications and costs just $50.
So what's new in SQL Server 2008 Express? Well it works with LINQ and the new ADO.NET Entity Framework. It also has the new FILESTREAM type, integrates with Microsoft 2007 Office System and has an integrated Full-text search. Interestingly you can also do spatial searches. You specify a polygon shape (in SQL), ie an area defined by a number of x,y points and then find all locations that intersect that. It's very useful (and fast) for proximity searching as in all points within a radius of 50 miles. I've added download links to all three versions on the Windows Tools and Utilities.