I am the author of the blog post mentioned by LouD above. I agree with those comments, that the price per unit of performance has increased quite significantly on Azure SQL DB under the new service tiers (i.e. Basic, Standard and Premium). If you have significant I/O requirements, the new tiers will likely prove expensive.
Also, just to point out however, that while there is no SQL Server License Cost for running SQL Express in a VM, remember that in a VM you do pay for the amount of I/O that you use, so lots of reading and writing increases the cost (where as Azure SQL DB is flat fee based). It is difficult to say whether the VM will be cheaper or more expensive than Azure SQL DB without knowing database size, I/O rate required, etc. But, finger in the air, I guess that for a small database, the VM may well be cheaper.
If you have multiple databases, that probably pushes things further in favour (cost wise) of the VM. At the moment there is no way to share resources across databases in Azure (e.g. you have five databases, four are quiet but one is busy, the busy one cannot use any of the resources allocated to the other four, those resources are essentially wasted). Microsoft have stated they are working on something that will make sharing resources / cost across groups of databases easier. They have committed to having this available before the end of Web/Business (which happens in September 2015) but other no details yet.
Also remember that Azure SQL DB and SQL Express in a VM don't offer the same set of features. The Database-as-a-service in Azure offers a whole lot more features, e.g. automated backup, more HA possibilities, managed servicing, etc. Also, and very important, note that Azure SQL Database supports many, but not all of the features of on-premise SQL Server. The online documentation provides a quite detailed breakdown of the on-premise features, SQL Statements, etc that are NOT supported on Azure.
Also, I would be cautious using the large table posted above by Vdub as a guide. It was taken from a 2012 blog post and much has changed since then in Azure SQL DB. The Azure SQL DB column in that table refers to the Web/Business editions that are being retired next year. Compared to the content of the table above, the new service tiers cost more (generally speaking), have additional HA features, don't support federations in the same way, have a higher uptime guarantee, etc.
Finally, just to follow on from a comment that LouD made... The new service tiers do support a limited amount of bursting. For example, see this chart:
in this post:
The start of this test (a sequential read from disk) does show a small increase / higher burst rate in the first few minutes. This is however, roughly speaking, less than that typically available in Web/Business.