Good question. You got one of the main differences right away. The initial upfront cost. CircleCI Convenience Images (the pre-built images you refer to) are designed to be a starting point for the majority of CircleCI users. They make it faster to get started building on CircleCI by providing the tools that most users need.
CircleCI Convenience Images
- Pros
- already made, faster to get started
- no Docker knowledge necessary
- more likely to be cached meaning faster builds
- changes with CircleCI platform requirements meaning always works on CircleCI
- Cons
- changes with CircleCI platform requirements meaning it can occasionally break your workflow
Private Images
- Pros
- only install what you need, light-weight efficient images
- can contain secrets & proprietary code (in the image, your code on CircleCI will always be private if you so choose)
- can be based on your Docker image used in production (if you do that), which is the best way to test your code
- Cons
- more work and Docker knowledge required
I'd also like to to mention that this question asks the trade offs between the CircleCI Convenience Images and private Docker images. The question fails to mention the third option. Any Docker image can be used with the CircleCI Docker executor, including other public images not published by CircleCI.
In terms of pros and cons, those images will share some of the pros and cons of both Convenience Images and private images. The key difference there is the trust factor. Knowing who publishes the image, their intentions, commitment to maintenance, etc should play a factor in your choice.
I hope that helps.