Project Online is the name or branding that Microsoft has created for their cloud based instance of Microsoft Project Server (hosted by Microsoft).
If you believe Microsoft Project Server is right for your organization, but are uncertain whether Project Online or Project On-Premise is the best choice; you might want to work with an expert.
There are many pluses and minuses of either approach. Some things to factor in; the size of your organization, your current project management maturity, how you intend to leverage the solution today as well as in the future (including custom reports), and in addition to this, one would need to understand the current capabilities of Project Online versus Project On-Premise. At the time of this article; there are a number of features Microsoft has been able to quickly implement into their Project Online solution, but have yet to incorporate into Project On-Premise.
Additional information on Project Online from these links:
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