Public or Private or Hybrid Clouds ?
Both Public and Private Clouds work best differently in specific individual organization usage scenarios depending on degree of optimization for cost (
public cloud ) or optimization for control (
private cloud) or a mix of the two (
hybrid cloud).
One definition of a cloud service is based on 3 criteria :

It is accessible via a Web browser

It has Zero Capital Expenditure to get started

It has a
Pay As You Go Model.
Private Clouds would not fit the definition above as they would have upfront organizational capital expenditure.
Organizations would go in for private Clouds when they desire the Cloud Computing benefits of Elasticity, Upward & Downward Scalability & Manageability for Computing, Storage & Bandwidth Resources within their own computing environments without the risks of the same with an external entity.
Microsoft describes 5 basic types of Workload Patterns that are Optimal for the Cloud :

On & Off Workloads (Batch Jobs)

Fast Growth Offerings

Unpredictable Bursting (Unexpected / Unplanned Demand Peaks)

Predictable Bursting (Periodic Seasonal Trends)

New IT Resource Based Product / Service Development.
For the five types of cloud work flow patterns, all the 3 possible Cloud options (Private, Public, Hybrid) could get used by customers based on need.

Factors that determine the same are as :
Organizational Size - Large Organizations would tend to use Private Clouds
more.
Medium Sized Organizations would use Hybrid Clouds and Small
Organizations & Individuals would use Public Clouds.
Ownership Of Organizations - Government & Publicly Owned Organizations &
Institutions would tend to use Private Clouds.
Others would use a mix of
Hybrid & Public Clouds.
Regulatory, Privacy & Governance Concerns & Constraints - Will determine
the location of hosting data and apps whether in an internal private cloud or
an external public cloud.
e.g. Health Data of EU Citizens can only be stored
within the EU geographies.
US laws such as
HIPAA,
Sarbanese Oxley & Credit
Card industry's
PCI standard impose stringent controls on internal personal
data.
They cannot be stored outside a physically visible organizational asset & boundary.
Capabilities of Existing IT Infrastructure - Organizations whose existing IT
Resources cannot fulfill the 5 possible optimal cloud work flow patterns would
look at cloud computing as an option.
The decision to go for any of the 3
cloud types (private, public & hybrid) would be based on many factors :

Hardware / Software Availability

Procurement & Provisioning Time lines

Present / Future Capacity Constraints

Disaster Planning & Recovery
Requirements

Availability / Costs of Real Estate

Electricity & IT
Administration Resources.
Degree of Optimization of Internal Organizational Capabilities - Organizations
may choose to outsource non-core capabilities and applications to highly specialized domain specific vendors (e.g. Email, CRM) who could provide these resources on public clouds at a much lower cost.
Internal core capabilities (e.g. sophisticated proprietary unique specialized applications) where control is paramount would be safe internally in a closed physical private cloud environment.
TCO calculations also would need to be done accurately to determine the best
choice.
Organizational Boundaries - Ability of applications to work across organizational boundaries & successfully
address security & integration concerns is also important.