The ways Dynamics CRM can help your business are ever expanding, and the Project Service Automation and Field Service solutions cover a large part of the CRM ecosystem that was previously not available. In order to better guide your business to the solution that meets your needs, some sample scenarios of their usage are provided below which will hopefully provide some guideance on which solution might suit your needs (or if you could benefit from both, which may be the case for many organizations).
Project Service Automation has been mentioned in other posts, but to make things easy, you can think of it as a way to manage projects (similar to what Microsoft Project can do) using tasks, resources, roles, and time tracking. It's targeted at services firms, who want to include not only products in their product catalog, but also services, which have rates, resources to execute the services, and projects to manage the delivery of services.
An end-to-end scenario for Project Service Automation might look something like this:

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Lead
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Inside Sales brings in a lead for an engagement requested by a customer. The lead is qualified by Inside Sales and an account, contact, and opportunity are generated. |

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Opportunity
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An Account Executive works with the customer to define the requirements for the engagement, which includes various service offerings your organization provides, along with the rates and costs for the services.
The opportunity can be added to your forecasting in CRM to be able to plan for possible resourcing needs in the near future, and to make any gaps more visible ensuring you can give customers an accurate timeline prior to them signing the engagement.
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Order
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When the opportunity is won, an order is generated based on the opportunity which includes all the service line-items. The order is sent off to the customer for execution via DocuSign, which is signed by the customer directly through the browser.
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Project
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A new project is created in Dynamics CRM, which includes milestones, tasks with prerequisites, alloted hours, and required roles. A Project Manager requests resources for the project, and a Resource Manager reviews the requests and assignes a resource(employee, consultant, etc.) to the project.
Resources execute the project, entering their time directly into CRM via the browser or mobile client (or Outlook in the future possibly), and the Project Manager monitors the remaining time and budget on the project using built in reporting.
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Archive
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Upon completion of the project, the milestones, tasks, timeline, and other features can be saved as a Template for future use. The resource utilization metrics, budget, actual costs, and other aspects of the engagement can be tracked to view overall performance metrics of your organization.
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Feedback
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Once the project has been completed, you can automatically send a survey using Voice of the Customer, ensuring continual monitoring of performance and constant improvement of service.
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This scenario is generic, but demonstrates how Project Service Automation fits into the overall use cases of Dynamics CRM. To be clear, this is not just a tool for service firms, but for any organization that wants to track projects or engagements in CRM. The projects do not need to be linked to Opportunities, and if desired, no other features of CRM need to be utilized besides Project Service Automation. It is truly a versatile and scalable product that can be used by almost every organization.
Jumping over to Field Service...
Field Service in Dynamics CRM can fit in with the above scenario as well, or be completely separate. It's targeted more at service organizations that need to track resources and schedule work with customers. The scenario above can actually be extended to match a need for Field Service as well, if we assume that the project engagement for the customer was to install some physical equipment which requires maintenance or repairs.

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Service Request
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A service request comes in from a customer and is added as a case to CRM. All relevant information on the case is added, entitlements are applied to determine SLAs and turnaround time. The functionality here is all out of the box Dynamics CRM. |

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Work Order
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It's determined that a company resource will be needed in the field to address the case, so a work order is created in Field Service, and associated to the case. If this is part of a larger issue or other cases are involved, multiple work orders can be associated with a case, or multiple cases with a work order.
The work order includes additional information such as the type of work, resources required (which may include people resources as well as tools, trucks, locations, or other 'things' with time, presence, or similar limitations). Field Service can be configured to automatically set the resources required based on the work requested, or this can be manually entered. SLAs may apply here as well, and can be tracked similarly to how they are with Cases.
In order to ensure the work order includes all information required to complete the work, customer assets, locations, contact, etc. can also be added.
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Schedule
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A single view can be utilized to select resources (left), see the current schedule (middle), and view unscheduled work orders (bottom). The work orders can be dragged to the schedule onto a specific resource, which allocates that resource for the time specified manually or the automatic time estimated from the work order. |

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Resource
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The dispatcher or group managing the schedule will select the resources for a work order, which many include the people equipment, locations, or other equipment required. Customer assets can also be included in the work order, ensuring communication with the customer is done ahead of time to make the assets available. |

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Dispatch
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Resources are dispatched to complete the work orders. They can utilize the Field Service Mobile App to see their schedule, get directions, review work orders, or handle other tasks in the field. Once completed, they can update the work orders directly from the field. |

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Schedule
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When work orders are updated, the schedule board instantly reflects the status, and those managing dispatch can continue scheduling additional work orders for resources and update customers as needed. |
Note that in this overall scenario, we could use Project Service Automation to manage a maintenance schedule for equipment installed for a customer, and use Field Service to dispatch the actual resources to perform the maintenance. This is a very simple use case for these two solutions which can be extended to meet the needs of just about any organization.
Hopefully one, or both, of these scenarios can be linked back to what your organization is currently doing, or looking to do in the future. Project Service Automation and Field Service will continue to grow into the CRM ecosystem and can be an integral part of any CRM platform.