It looks like Project - View and Project - View management Access are working as same when the user has Resource - Hard book access to a user.
The description of the access right Resource - Hard Book - All in Clarity states:
"Allows user to Soft or Hard book all resources to an investment that the user has been granted view management rights."
Based on that description should both Project - View and Project - View Management allow the user to update the team on a project?
Release : All Supported Releases
Component : CLARITY RESOURCE MANAGEMENT
Both Project View rights will give the user access if they have Resource - Hard Book or Resource Manager (auto) access to the user:
The description only mentions Project - View Management as it's the one access right available at the global level (Project View is not).
See Project Access Rights for more details on these two project view rights and other access permissions.