Windows Azure SQL Database is a Platform-as-a-Service (PaaS) offering of Microsoft's SQL Server. Paas is different from physical or virtual machine instances of SQL Server. Those other options would be referred to as infrastructure. If those virtual machine instances of SQL server are hosted by a cloud provider such as Microsoft Azure, then those infrastructure offerings become known as Infrastructure-as-a-Service.
PaaS options would be attractive to businesses who do not have the resources (time, money, or people) to manage a large infrastructure. PaaS offerings eliminate the capital expenditures associated with on-premise infrastructure. They also minimize the operating expenses associated with infrastructure or IaaS, since the maintenance costs of Patching and upgrading your SQL Server instances is taken care of for you. Even backups and high-availability options are enabled with the push of a button if desired.
There are still some operating expenses associated with Paas. Above the actual fees charged for Windows Azure SQL Database, companies can still benefit from the Database Administrator's (DBA's) skillset. Index maintenance, complex High Availability (HA) or Disaster Recovery (DR) plans can still benefit from DBA oversight.
It seems that DTU's are not completly comparable between service layers. When moving a workload of 95 DTU from a S3 database (100 DTU) to a P1 database (125 DUT) one can expect to see the average load on the P1 to drop below the 95 DTU.
There is no official public statement from Microsoft on why they choose to use DTU's as the measure of performance/scaling for Azure SQL DB. However, this is in line with other services like Stream Analytics and the Event Hub who's performance is also measured in custom metrics, e.g. Streaming Units and Throughput Units.
There are also eDTUs, which are related to the use of Elastic Pools. Performance wise a DTU and eDTU are the same.