Ably Realtime and Amazon SNS solve fairly exclusive subset of problems:

Pub/Sub - the primary difference

Ably comes with a variety of client SDKs that you can use to connect your clients directly to Ably's Realtime platform. Ably then fans out the published messages to all the relevant subscribers - when using the general Pub/Sub via channels.

Whereas, Amazon SNS is mostly used for server to server operations while email and SMS being the only two ways in which these message can reach the end clients. SNS let's your AWS services, such as Amazon EC2, Amazon S3 and Amazon CloudWatch, to publish messages to certain SNS topics to trigger other AWS services such as Lambda, SQS etc. 

Although, topics in SNS are analogous to channels in Ably, they are both used for different purposes as explained above.

Other realtime features

Apart from simple Pub/Sub - of which Push Notifications is one part, Amazon SNS does not offer any other realtime features such as Presence, Firehose, Message Queues, etc which are directly pluggable to your client applications and/or your servers.