There is some confusion between CDI and CDP primarily because a CDP is essentially a CDI that also has identity resolution and audience building capabilities.
The core premise of a CDI is to track customer data from first-party data sources – your website and apps – and sync the data to a data warehouse as well as to third-party tools (downstream systems where data is eventually activated).
Segment, through its range of products, makes it easy to understand the differences between CDI and CDP. Connections, Segment’s core product, is a CDI solution whereas Personas is Segment’s CDP offering that is sold as an add-on.
Other CDI solutions include RudderStack, MetaRouter, Iteratively, and Avo to name a few. They all have different capabilities and support different destinations but all of them can be used to track data and sync it to downstream tools where data is activated.
CDPs like mParticle, Lytics, Tealium and others bundle a CDI solution with CDP capabilities that enable building and syncing audiences to specific destinations rather than only syncing raw customer data which is typically the case with a CDI.
To summarize, both CDI and CDP solutions enable data activation by syncing data to downstream systems like sales, marketing, advertising, and support tools where data is activated.
CDPs, however, are preferred by less-technical folks as they are able to build and sync audiences to downstream tools using a visual interface offered by every CDP. And doing so enables them to move fast without relying on engineering or data teams.
CDPs are Going Beyond Infrastructure
It’s worth mentioning that while CDP vendors have largely been focused on collecting and moving data, some of them also allow you to activate the data and orchestrate campaigns across multiple channels such as email and SMS.
Exponea is one such CDP that has inbuilt functionality to build personalized, data-led experiences. After its acquisition by Twilio, Segment is moving in this direction as well and already integrates deeply with Twilio for SMS and Twilio-owned SendGrid for emails. 
This is a natural expansion for CDPs and I believe more vendors will either build or buy activation products going forward.