This is indeed a discussion and is non-trivial. There are some important things that come to mind right off - Enterprise provides a higher level of security than AGOL does because it essentially operates behind your organizations firewall, the other has to to do with using ArcGIS Server to create services (the REST service you mention) that are consumed in a web browser in some fashion be it through a web map or some other query that returns a result. This is largely what the hosted feature layers you create in AGOL are, but in Enterprise they are hosted 'locally' (whatever that means in this day of Cloud based architecture) rather than in AGOL's cloud. Managing REST services is a bit of an arcane art that takes training if you haven't used them previously - its easier now than even a few years ago, but its still a different thing than just publishing a layer to AGOL.
One advantage that Enterprise offers is that you aren't paying for storage as you are with hosted services and other resources in AGOL - this can be an issue if you find yourself paying for a lot of storage (using Service Credits) in AGOL, so you'll want to do an analysis of what you are storing currently, how that is likely to change, and what the impact of large increases in your data are likely to mean.
While it sounds great to go Enterprise, I think there is a big hardware/network and infrastructure hurdle that a lot of organizations just don't have the technical chops to manage, so you really need to involve your IT section in planning this. I think if you are doing a lot databases that require multi-editor workflows, or you have very large databases you are hosting, then Enterprise is probably the way to go. Understand too that features available in AGOL are not 100% compatible with Enterprise, at least not when those are released, so if you want to make use of some new fancy capability offered with the most recent update of AGOL, that isn't going to be available to your Enterprise right away, there is always a lag between those features being released and then updating to Enterprise.
Some planning issues you have to understand: can your org support the necessary IT infrastructure (servers, storage, network capacity, on-prem versus cloud, etc.) and technical management that Enterprise requires? Do you need to maintain an AGOL organization for outside collaboration or more public sharing? What are your security needs? How are your GIS data currently stored and managed, how will Enterprise change this? What are your current day to day workflows and will those translate to an Enterprise environment. Will you be able to migrate existing AGOL layers/maps/apps to portal and how much time ($$) will it take to manually reproduce those items you can't migrate? Ultimately, you're going to have to sit down with ESRI (or even 3rd party vendor) to help plan this out and you'll end up talking about an Enterprise License Agreement which is a different level of hassle than the already complicated licensing ecosystem involving pro and AGOL.
