Apposition
to add explanatory or descriptive words immediately after prior words, separated by commas; from the Latin ad 'near' and positio 'placement'; added elements are called 'appositives'
Parenthesis
to insert into sentences expansive or digressionary words that interrupt normal syntactic flow; from the Greek para 'beside' and thesis 'placing'; added elements are called 'parenthetical expressions'

Examples

Apposition

Always taciturn, Angus McAllister merely grunted.

E.J. Smith, captain of RMS Titanic, previously commanded RMS Olympic, the word's largest passenger ship before Titanic.

Vladimir Nabokov
Lolita, light of my life, fire of my loins

Truman Capote
The village of Holcomb stands on the high wheat plains of western Kansas, a lonesome area that other Kansans call 'out there.'
Parenthesis

We should, I suppose, have made a booking.

The bonus was surprising but moderate ($500).

Let me say - this may surprise you - I've never trusted her.

Abraham Lincoln
The mystic chords of memory will swell when again touched, as surely they will be, by the better angels of our nature.
↓
'[T]he better angels of our nature' is one of Lincoln's most memorable phrases.

Purpose

Adding Information Types
Apposition and parenthesis add further information to support sentences. They are grammatical schemes of composition and artful schemes of rhetoric. Apposition seldom appears in speech, and parenthetical speech requires signalling, emphasis and pausing. Grammarians distinguish between apposition and parenthesis to a greater extent.
Differences
Apposition
Parenthesis
  • more informational than rhetorical: supplies more important information like definitions, categorizations, descriptions
  • less syntactically interruptive
  • more grammatical: appositives are coordinate structures
  • commoner in writing than in speech
  • more rhetorical than informational: supplies less important and tangential information like the author's commentary
  • more syntactically interruptive
  • less grammatical: parenthetical elements 'break away'
  • commoner in speech than in writing

Usage

Apposition and parenthesis are commonly covered in composition courses so we probably don't consider them rhetorical schemes. The following general advice is offered.
Appositive and parenthetical elements add extra information that disrupts sentences. We must therefore:
  • set off the extra information with punctuation symbols
  • consider whether the information is necessary - removing it has no effect on sentence grammar
  • weigh the cost of adding the information against the weight of interrupting the reader
Punctuating Appositives & Parentheses
Set off parenthetical elements with dashes or brackets (parentheses).

Einstein – I mean my pet dog, not the scientist – gazed at me questioningly.

Brackets are more commonly used outside of literature, especially in academic writing, because they compactly enclose additional information.

The American festival of Halloween (31 October) is not celebrated in many countries.

Homeopathy cannot be considered a medicine unless one accepts its 'theory' (that water has 'memory') is scientific.

Alchemy (a medieval precursor of chemistry that believed base metals could be turned into gold) was widely practised until as late as the 18th century – Isaac Newton practised much more alchemy than physics.

Grammatical errors in originals are indicated using [sic] in square brackets after the error.

She said, 'Between you and I [sic], he's unlikely to even get an interview.'

A common appositives error is to forget the final comma. This may confuse readers. Compare:

Sally, the club president and her partner opened the ceremony. (Did three people open the ceremony?)

Sally, the club president, and her partner opened the ceremony. (Sally opened the ceremony with her partner.)

Appositives Replace & Combine
Use appositives to economically replace subordinate clauses and combine sentences.

Relative Clause:           Lisa, who is a dear friend and lives next door, was born in Denmark.

Multiple Sentences:    Lisa was born in Denmark. She is a dear friend. She lives next door.

Appositive:                   Lisa, a dear friend from Denmark, lives next door.

Appositives as Adjectives
Use appositives to combine noun groups. Or use them as adjectives.

Multiple Sentences:    The boat shed was weather-beaten but solid. We sheltered in it from the storm.

Appositive:                   Weather-beaten but solid, the boat shed sheltered us from the storm.

Appositives for Variety
Use appositives to create sentence-pattern variety. We can place them at the start, middle and end of sentences.

Start:                             That fearsome Aberdeen terrier, Angus, let out a chilling growl.

Middle:                        Angus, that fearsome Aberdeen terrier, let out a chilling growl.

End:                              A chilling growl was let out by Angus, that fearsome Aberdeen terrier.

Appositives & Parentheses Interrupt
Use parentheses sparingly because they are cognitively harder to process and slow readers down.

Necessary?:                  On the Origin of Species, Darwin's theory of natural selection, was first published in 1859 and went on to be republished in five further editions in Darwin's lifetime.

The author is probably well known by the intended audience. And if we must add extra information, less interruptive means of adding are available.

Alternative 1:               Darwin's On the Origin of Species was first published in 1859 and went on to be republished in five further editions during his lifetime.

Alternative 2:               Darwin's theory of natural selection was first published as On the Origin of Species in 1859 and went on to be republished in five further editions during his lifetime.