I started the novel but I’m going to reduce it down - hopefully.
Active Directory is an on-prem authentication and authorization service released in 2000 and is tightly integrated with Windows clients and servers. It does not work with devices that run iOS, Android, Linux, or MacOS. It organizes users and computers into groups, organizational units, domains, and forests. All computers within a domain have an implicit trust relationship with each other. Networking was new at the time and MS wanted to make sharing easy. This was great pre-Internet but is now more of a security liability. AD uses a single factor for access: the domain\userID and a secret (password). MFA can be added through 3rd parties like DUO.
Azure Active Directory is neither Azure nor Active Directory because naming things is hard. It is a hosted authentication and authorization platform based on OpenID Connect which provides an authentication component to the authorization features of OAuth 2.0. AAD works on all web platforms regardless of operating system and comes in a free tier, an M365 tier, and two premium tiers with additional capabilities. Azure AD is flat and not like a tree. There is little implicit trust. Access is granted explicitly. In addition to users, computers, and groups, AAD also registers applications. We see this when we set up Kinetic, which can be on-prem or in the cloud like other services: M365, GitHub, SmartSheet, KnowB4, and many, many others. This means that if you terminate an account, all of the services are shut-down at the same time - true single sign-on across all operating systems and not just Windows no matter where you log in - and no VPN required. Registered applications can give API access to other applications. This means that one application can display information from others without breaking context: see emails from O365 in Kinetic, view Kinetic data in a Teams app or a custom web app that pulls in data from an external warehouse application. There are more web applications that can use AAD than Windows Auth which is only possible on IIS.
AAD also comes with multi-factor authentication right out of the box - including the more secure phishing resistant number-entry and passwordless options.
With AAD, one can also use Conditional Access rules. For example, if a user logged into the home office and an hour later a login attempt comes in from Asia, it can require an extra MFA to continue or block the request outright. If a global admin tries to log in, a rule can indicate that the login must come from a small set of devices. AAD can also require that the device has a certain level of health checks approved like Windows Updates are applied, Endpoint security running, etc. before granting access.
Finally, with a single login system, there are fewer passwords for users to remember or get compromised. That also gives better forensic logging since AAD logging is far more difficult to alter or delete than on-prem logs.
This is what you might be missing. That and breakfast with my family.