Really DynamoDB and MySQL are apples and oranges. DynamoDB is a NoSQL storage layer while MySQL is used for relational storage. You should pick what to use based on the actual needs of your application. In fact, some applications might be well served by using both.
If, for example, you are storing data that does not lend itself well to a relational schema (tree structures, schema-less JSON representations, etc.) that can be looked up against a single key or a key/range combination then DynamoDB (or some other NoSQL store) would likely be your best bet.
If you have a well-defined schema for your data that can fit well in a relational structure and you need the flexibility to query the data in a number of different ways (adding indexes as necessary of course), then RDS might be a better solution.
The main benefit for using DynamoDB as a NoSQL store is that you get guaranteed read/write throughput at whatever level you require without having to worry about managing a clustered data store. So if your application requires 1000 reads/writes per second, you can just provision your DynamoDB table for that level of throughput and not really have to worry about the underlying infrastructure.
RDS has much of the same benefit of not having to worry about the infrastructure itself, however if you end up needing to do a significant number of writes to the point where the largest instance size will no longer keep up, you are kind of left without options (you can scale horizontally for reads using read replicas).
Updated note: DynamoDb does now support global secondary indexing, so you do now have the capability to perform optimized lookups on data fields other than the hash or combination of hash and range keys.
