The AS-ROTK tells us that these two ancient wind instruments were chosen to represent "the most ancient and revered kingdom of Men."... Gondor

Pan Flute

The pan flute, alternately called pan pipes, consist of a number of tuned pipes of graduated lengths which are bound together and blown across the top. The breath is blown horizontally across the open of the pipes, which are stopped at the other end. Each pipe gives out one note, but by overblowing, that is, increased pressure of breath and tension of lips, harmonics are produced. They are used in many different cultures.

USE IN SOUNDTRACK

Although Doug Adams describes the pan flutes (and the wood flute) as being associated with Gondor, they are more specifically associated with Faramir. But maybe Faramir is the better representation of Gondor than his father ... at least until a better representation comes along. HS COMMENTS (HS An Introspective)

The pan flutes hint at the Faramir and Denethor theme as Faramir says what's been left unsaid, Denethor wishes Faramir had died in place of Boromir. As Faramir rides from Minas Tirith to do his father's bidding, the pan flutes take up that theme in earnest. 

ARTIST

Ulrich Herkenhoff played the pan flute in ROTK.

1. Sample from CR-ROTK, Disc 2, Track 3, Allegiance to Denethor, 2:21-2:45

Wood Flute

Wood flutes can be keyed or not and are often referred to as Irish flutes.

USE IN SOUNDTRACK

See my comments above about these instruments, Gondor and Faramir.

The wood flute seems to have been used exclusively for the scene in the Houses of Healing where Faramir comes to Éowyn and, in talking together, each finds their sorrow eased a little.

ARTISTS

Stewart McIlwham played wood flute for ROTK when Faramir and Éowyn stand together in the Houses of Healing.

1. Sample from CR-ROTK, Disc 3, Track 13, The Last Debate, 1:08-1:58