CDK maintains the state with AWS itself, Pulumi operates their own state service (with option of self management). I have never used CDK, but I've heard that it's hard to recover from state corruption in CDK.
This is noteworthy because many times CDK's constraints match CloudFormation's constraints. I've noticed a few issues in the CDK github repo that a feature is incomplete because they are waiting for a change or feature to land in CloudFormation first.
I never have to worry about losing the state or ending with the incomplete state, and no matter what, I can always just delete the whole CF stack (in rare cases, you might have to retry deletion, but it never loses resources.)
That said, I don’t think that “generates CloudFormation JSON” is a bad attribute of CDK, but it is a major difference between Pulumi and CDK. In particular it means arbitrary values cannot be used for control flow at run time with CDK, since the template has a static generation step. However, it also does not require an additional tool at runtime, which has value in some cases.