A Service Desk/Help Desk is a primary IT function within the discipline of IT Service Management (ITSM) as defined by the Information Technology Infrastructure Library (ITIL).

It is intended to provide a Single Point of Contact (“SPOC”) to meet the communication needs of both users and IT staff  and also to satisfy both Customer and IT Provider objectives.
The IT Help Desk is viewed as more tactical or I’ll say more day to day actionable helping to quickly resolve end customers immediate needs and technical issues and incidents (sometimes referred to as externally focused).  It can be separate or part of the Service Desk to improve the overall organization’s Customer Services. The ultimate goal of the Help Desk is to offer first contact resolution as efficiently and quickly as possible.
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The IT Service Desk is thought of as a broader term that is more strategic and cross organizational (sometimes referred to as internally or organizationally focused).  This looks at the business needs rather than solely focused on resolving the user’s needs and takes into account the broader business context. The ITIL 3 definition of the Service Desk (Service Operation) is the Single Point of Contact between the Service Provider and the Users.

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Can a company just have a Help Desk and not a Service Desk and vice versa?

There are no specific rules  and there are  instances where a company does not require or may not be ready for the processes and service offerings of the broader Service Desk.  In that situation a Help Desk is perfectly fine to meet these tactical needs.  The Help Desk provides business users a place to contact when they are having specific IT issues.  The goal here is to have that initial point of contact to reduce the wait time and get the issues resolved quickly to get the user working and productive. The Service Desk looks at the overall IT processes and the individual components that function and interact with each other on both a software and process level.   One area that is a must in any IT service offering is the ability to manage specific end user problems and issues.  A Service Desk either has to have Help Desk functionality embedded as part of the Service Desk infrastructure or has the ability to link into a Help Desk offering for the end user.  The Service Desk is typically focused on corporate strategy and ensuring all the IT functions are currently working and will work in the future, they must also have the ability to ensure all end users up and functioning.
Some additional and important capabilities for a Help Desk and Service Desk deployment:

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