Windows Azure's SQL Database is a relational database, with all the things you'd expect from a relational database (multiple indexes, stored procedures, powerful queries, etc.). Azure Table Storage is a non-relational, massively scalable (up to 100TB per account) storage facility, where entities are located by partition key (a colocation of entities) and row key.
If you want to have a very simple storage mechanism that doesn't require sophisticated relational operations, Azure Table Storage will work quite nicely.
EDIT June 7, 2012: Updated with Spring Release pricing
There are cost differences too. SQL Database starts at $4.99 for 100MB, scaling up on a tiered scale (about $26 for 5GB, $125 for 50GB, $225 FOR 150GB) but has no transactional costs. Azure Table Storage runs $0.125 / GB (or 0.09 / GB without geo-replication), dropping in per-GB price as quantity goes up, but has a $0.01 / 100,000 transaction cost (nominal for low volume systems, but could be significant with very high volume systems). Full pricing details are here.
There's a fairly recent article in MSDN Magazine that goes into greater detail regarding use cases, differences, etc.
If you're going for a straightforward migration of what you have in place today, SQL Database will closely match what you have in SQL Server Express 2005. However, since it's only a prototype at this point, it's worth re-evaluating your needs.