As commented by Mike, manage.windowsazure.com is the current production Azure Portal while portal.azure.com is the preview portal which will eventually replace the production portal.
From an underlying technology perspective, there's one big difference between the production and preview portal. Production portal makes use of Azure Service Management API while the Preview portal makes use of Azure Resource Manager (ARM). Along with ARM API, you get Role-based access control (RBAC) that enables you to grant granular permissions on your Azure resources to your team members. In the production portal, there's only a concept of Subscription Administrator and Subscription Co-Administrator.
Not all services in Azure has been ported to make use of ARM API as of today and that's why you see only few services in the preview portal. Services that make use of ARM API (all the new services) will only show up the preview portal.
As to when to use what portal, just see the Azure services you need to manage. Based on how they can be managed, you will choose between production and preview portal. Also please note that functionality for a service may differ between portals even though it is present in both portals. That may be another criteria between choosing the portal.
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