Identity Access Management is what they call the Role-Based Access Control system in Azure subscriptions. Basically, it allows you to give users certain roles on subscriptions, resource groups, or individual resources.
Azure AD is a more general identity management solution. It allows you to manage users and applications, users’ access to those applications and more. Calling it the “same as on-prem AD” is not really quite right, since their features differ quite a lot. The general purpose is similar to on-prem AD of course.
Logically Azure subscriptions exist within one Azure AD (they are linked to one). But having your user account in Azure AD does not give you access to Azure subscriptions! In addition to a user having to be a member (or invited guest user) of the AAD assigned to the subscription, you also have to use the RBAC (IAM) system to assign roles for users.