Difference between Chatter API and Connect API in Salesforce?

Chatter API is REST API for Chatter to display Salesforce data, especially in mobile applications. Responses are localized, structured for presentation, and can be filtered to contain only what the app needs.

Connect API provides apex classes for accessing the same data available in Chatter REST API. Use Chatter in Apex to create custom Chatter experiences in Salesforce.

When to Use Chatter REST API
Is Chatter REST API the right API for you?
Use Chatter REST API to:Build a mobile app.
Integrate a third-party web application with Salesforce so it can notify groups of users about events.
Display a feed on an external system, such as an intranet site, after users are authenticated.
Make feeds actionable and integrated with third-party sites. For example, an app that posts a Chatter item to Twitter whenever the post includes #tweet hashtag.
Create simple games that interact with the feed for notifications.
Creating a custom, branded skin for Chatter for your organization.
How Is Chatter REST API Different from Other Salesforce APIs?
The following are some of the differences between Chatter REST API and REST API:

Data is structured for rendering on websites and mobile devices.
Returned information is localized to the user’s time zone and language.
Changed values that are tracked in a feed are returned as value-pair representations.
Rate limiting for Chatter REST API is per user, per application, per hour. The rate limiting for SOAP API and REST API is by organization.
If needed, extract an ID from the REST API and use it in requests to Chatter REST API.
Chatter in Apex
Many Chatter REST API resource actions are exposed as static methods on Apex classes in the ConnectApi namespace. This namespace is also referred to as Chatter in Apex. Use Chatter in Apex to develop Chatter applications on the Lightning Platform without using HTTP callouts from Apex.