What's the key difference between the Office 365 E1 and E3 plans?

With the E1 subscription you don't get the office client, just the browser office web apps. You can however use an existing version of office. The minimum supported version is Office 2010 with some reduced functionality compared to Office 2013 and 2016 (Office 2016 has just been released).

E3 Subscriptions provide continuous automatic updates to the latest office version. The E3 also supports dashboarding Infopath Web Forms and Services such as Vision Services, Excel Services and Access Services.

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