Exchange ActiveSync is a protocol developed specifically to support mobile devices that are written to use the ActiveSync API.
EWS is a collection of several different web service APIs to support a multitude of services. Some of these are:
Outlook for MAC
Availability
Mailbox Replication Service Proxy
Also, many third-party vendors will use the EWS APIs to develop their products as it gives a robust platform for accessing mailboxes as the mailbox owner. A prime example of this is many archiving solutions.