Push messages are short text messages (140 characters), that you can immediately and directly send to the users of an app. They appear as short messages on the smartphone, even when the app is not currently active.
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As soon as your users have clicked the push notification via the lockscreen of their phone, the message disappears.

Users need to accept and activate push notifications via iOS or Android before they are able to receive them. 

Info
Push messages are not sent to iOS Web Apps, they can only be received by Android Web Apps.
Via the native app, push notifications can be received on both iOS and Android. 

Push messages can only be send to users and not to visitors.
Visitors have only opened the app once without showing any further interaction.
Users have interacted with your app, changed pages or logged in. 

Chat messages can be send to all users or a group of users. Users can answer to your chat messages or can send a chat message by themselves to you.
You have the possibility to activate or deactivate chat messages.
That can be done in the general settings of your branded app.
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A chat message to all users can be send by clicking on Integrations > Branded App > Manage settings. Here you can click on the little tick box on top of the user list and then you can click on the button "chat message". Now you can write a message and send it to all of your app users.

If you want to send a chat message to a group of users, you can do this by clicking on Integrations > Branded App > Manage settings and selecting the users that should be included, by ticking the check boxes in front of their names.
Then again you can click the "chat message" button. You can see the number of people the message will be send to. 

If you want to send a chat message to just one user, you can also click on Integrations > Branded App > Manage settings. Here you can find little chat icons in the row of the specific user. If you click that one the chat window with this specific user will open.
You can send an individual message to just that one user. You can also see the messages that this user has send to you.