Unless you need a very simple and small database, Aurora is likely the better choice over MySQL when using AWS, for these reasons:
Aurora is compatible with MySQL. This means your existing code and MySQL tools will all likely just work with it as if it is MySQL.
MySQL may be slightly cheaper / run on lower spec VM's, but Amazon claim any extra cost for Aurora is more than compensated by the better performance (x5 performance claimed, e.g. you would spend more on compute resources for RDS MySQL to get equivalent performance). For very simple DB applications MySQL may be the cheaper choice, but for for more demanding applications Aurora should be better, potentially a lot better when dealing with lots of data, multiple instances and/or multiple regions.
Can be replicated with up to 15 read replicas of the database to increase read capacity.
Aurora is enterprise-class database designed by AWS but without the cost of commercial databases.
Most of the smarts is in the storage side of things. Its built specifically for the AWS storage infrastructure, whereas other database engines typically treat that storage as a traditional file system.
Aurora replicates each chunk of a database volume six ways across three Availability Zones. This replication is bundled into the price.
How MySQL compatible?
Most of the code, applications, drivers and tools you already use today with MySQL databases can be used with Aurora with little or no change. The Amazon Aurora database engine is designed to be wire-compatible with MySQL 5.6 using the InnoDB storage engine. Certain MySQL features like the MyISAM storage engine are not available with Amazon Aurora.