What is the difference between a "request" and an "inquiry"?

As a
noun, the word "inquiry" is commonly defined as an act by which information is sought.
In this instance, it is used in the form of a question or a query.  Furthermore, the word can
also refer to a formal or official investigation. Synonyms in this regard would be
investigation, examination, probe, review, exploration, analysis, review, hearing, and
inquest.

As a verb, the appropriate form is "inquire" and it has
the same meanings as defined above. 

A request (in its noun form) is defined
as an act in which one asks for something to be done or given, mostly as a favor, solicitation,
or petition. Synonyms in this regard would be plea, petition, application, call, demand, appeal,
and entreaty.

If used as a verb, it means to politely or formally ask for
something.

The difference between the two words should be clear from the
above definitions. To further clarify the contrast, one may look at the context in which each
word is used, as in the following examples:

Inquiry:

The committee lodged an
inquiry in which it sought more information on what it believed was excessive
expenditure.

Upon my inquiry about her whereabouts, she explained that she
had been to the grocer.

A medical inquiry has a very formalized
structure.

At his inquiry, Peter informed them that he lived in Aberdeen and
that he had been unemployed. 

I instituted an inquiry to determine the facts
of the matter.

You should inquire why the task was not completed.

I inquired why such a large batch of documents had to be completed.

Request:

The teacher's request for more
funding was turned down by the board.

I rolled down the car window at my
passenger's request.

The agreement was signed after the president's request
that they come to some kind of accord.

The student was told that he should
request permission to leave the classroom.

"I hope I have fulfilled your
request for greater care," said Norman.

The community lodged a request
with the council for greater protection.