I do not understand why Esri has two products serving similar purpose.
Why one should have ArcGIS Server and why one should choose ArcGIS Online?
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ArcGIS Online is Esri's Cloud platform. Under the hood it is powered by ArcGIS Server and allows organisations to benefit from utilising spatial information "in the cloud". In essence it would allow you to benefit from ArcGIS Server without having to worry about the implementation, deployment, and management overheads. It also provides online web mapping sites without the need for developing your own and basic "canned" GeoProcessing tasks.
ArcGIS Server is the core platform that powers ArcGIS Online. You would utilise ArcGIS Server when ArcGIS Online is not appropriate for your needs. For example, you might have very strict data security requirements that would prevent you from utilising a cloud service running on a shared infrastructure. Or perhaps you want to have complete control over your infrastructure and platform. Or maybe you want to create your own custom web site and services. As pointed out by D.E.Wright in his comment you would also choose ArcGIS Server when you want to create complex GeoProcessing tasks and implement custom workflows.
Ultimately it comes down to your specific needs. If you want to benefit from storing and accessing your spatial data online with ease and don't want, or don't have the skills, to manage your own server infrastructure then ArcGIS Online may well be appropriate. If you have specific data security requirements, or simply want full control over the deployment and management of servers then implementing your own ArcGIS Server environment would be appropriate.
In my view Esri has the right strategy offering both platforms as they serve different use cases and user types.