Need for push notifications
Especially on iOS you don't have a choice but to use their push notifications service APNS. There is no other way to receive notifications immediately because iOS may kill or neglect the TCP connections of your background app.
On Android it seems possible to use your own background TCP connection to avoid having to use push notifications. But you may still consider the use of the push notifications through GCM for the sake of improved battery usage.
Pricing
Neither APNS (iOS) nor GCM (Android) charge you for the service and you are allowed to send an unlimited number of messages.
Limitations and Requirements
Both services will delete message, when there are too many messages accumulated in the queue for an offline device - which makes sense because there is no point in delivering those messages hours later. You have to take that in to account, when writing your app (just do a poll when going back online).
Depending on the app you are writing, there may also be privacy concerns. Even if you encrypt the message itself, at least Apple/Google know when a notification is sent to a certain device, which may be a deal-breaker for certain high-security applications.
You will also need a server that is able to communicate to both APNS and GCM. There are open source solutions for that (e.g. easyApns for iOS and python-gcm for Android), but how easy their integration is depends on your server and the language it is written in.