Dynamics NAV and Business Central 365 are, under the hood, the same machine. First released in 1995, the on-premise Dynamics NAV came first by over 20 years. The cloud version, Dynamics Business Central 365, was only released in early 2018 but, because Dynamics 365 is built on Dynamics NAV, it takes advantage of much of that decades long time span of NAV development.
Features. There is some difference here.
Accessibility. LAN accessibility is really only applicable to on-premise. Both NAV and 365 offer WAN accessibility but in different ways. The cloud can only be accessed through a Web Client in the form of a web browser, be it Internet Explorer, Firefox, Safari or another. On premise gives you both a Windows and Web Client and yes, the Windows client will work from anywhere just as efficiently. NAV pages can also be plugged into SharePoint.
Multiple databases. Because NAV is a world class ERP it is released in over 40 regional versions, each with functionality built in for the legal accounting requirements of that region, be it Europe, Mexico or India. If you plan on using NAV in more than one region, you will need to run more than one database. Whereas this is easy to do on-premise, you would need multiple separate cloud accounts, each in its own separate regional datacenter, to do this in 365.
Data Warehousing. If you want to run analytics using a Data Warehouse or other application that combines data from multiple sources this would not be possible with a cloud application. With NAV on-premise not only do you have your database in a known location, this can also be combined with other data on your own file servers.
Programmability. Many companies choose NAV because if its open codebase that permits one of the friendliest development environments in the ERP world. With the on-premise version customizations can be done at any level of coding. Dynamics 365 modifications are limited to the “extension” model that limits modification to specific points of the codebase. Further, extensions must be packaged up in order to install them. Finally, because of security concerns, 365 does not allow customization that use .Net. In short, 365 is really not meant for anything other than light development and packaged add-ons. This may be well enough for some, but certainly not for all.
Cost of Ownership. Dynamics 365 is a subscription model, paid monthly, with no prospect of future acquisition. With NAV on-premise the product is fully owned upon purchase and the license does not expire. However, on-premise NAV does come with a 16% annual enhancement fee that in the end has you re-buying the product every six and a half years. Although this fee is not mandatory, if not paid you are locked out of adding users or add-on products to the system.