I think that the previous thoughts are very lucid.  I would like to echo the idea that there is a different writer's voice that is adopted in each setting.  For example, the letter can be personal and subjective whereas this voice might not be best for an expository or persuasive essay, which is more informational based, in general.  I do think that this voice helps to change the scope of each writing sample, whereas one can embrace emotions and sentimentality, while the other forums are more along the lines of seeing the subjective as part of something larger and the minimization of its use is done in large part to feed a more totalizing or objective part.

It all depends on audience and the type of letter or essay.  A letter to a familiar audience is much different than an expository essay, but a formal letter to an unknown audience will not differ greatly.

1.  Informal Letter vs. Expository Essay: a familiar letter is much more intimate in voice than an expository essay.  A writer of an informal letter to a familiar audience uses "sweet style": 2nd person, colloquial language, casual diction, perhaps humor.  It is likely filled with pathos, emotional language--perhaps laced with passionate love.  It is non-academic and has no formal claim or grounds with which to influence or change the audience's position.

2.  Formal Letter vs. Expository Essay: these may both be "stuffy" in style, which means they have theses (claims and grounds used to persuade).  This type of writing is academic and formal, highly organized (topic sentences, quotes, examples), and it relies on formal diction (3rd person).  It is likely filled with logos, logical and objective argumentation.  This is the language of academic discourse (high school and college).  Even though it is a letter, its purposes may be very similar to an essay.  Many "open letters" are meant to persuade a wide audience.