When comparing Conda vs Flatpak, the Slant community recommends Flatpak for most people. In the question“What are the best Linux package managers?” Flatpak is ranked 9th while Conda is ranked 13th. The most important reason people chose Flatpak is:
You can install flatpak packages on any distro you want.
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Pros

Pro Binary installs
They are fast and reliable as they do not need to compile before installation.

Pro Allows for multiple environments
It is great for developers since you can easily switch between complete environments with different versions of packages, for testing and development.

Pro Open source
Conda is open source and on Github, so if you see something wrong you can fix it and submit a patch.
Pro Cross-distribution
You can install flatpak packages on any distro you want.
Pro Application sandboxing
All applications are limited to a set of predefined permissions, enhancing privacy and security.
Pro A well-written documentation
Pro Flexible runtime management
You can install a lot of runtimes for different apps, making applications a lot more compatible while still allowing some applications to share their runtimes.
Cons

Con Doesn't have everything
Conda is relatively new and has a smaller user-base, so the set of packages available is limited.
Con No way to resume downloads
Any download that is canceled or interrupted will have to be started over from the beginning as there is no built in solution for resuming downloads.
Con Bloated
Due to the way Flatpack handles packaging, this can lead to a large cache being created which quickly inflates to unreasonable sizes. Not only this, but using flatpack requires a large chunk of space to be reserved for it's own file hierarchy.