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In a marching band I noticed there's three main components: percussion, woodwinds and brass. I was wondering what the relationship between brass (eg. trumpet) and woodwinds (eg clarinet) in a march. They seem to be both melodic instruments. Do they play the exact same voicing? Or does one harmonize the other in some way?

Seems like you know piano, so maybe this will clarify:

Imagine each instrument in the band corresponds to one of your fingers. When you play a big chord, you might use most of your fingers to play it, and each finger plays a different note. All the fingers together make the chord.

That's how bands and orchestras work (sort of). And the different instruments might be more often used for specific fingers. Like the tubas are more like your left hand fifth finger. They usually are playing the lowest notes. Piccolo and flutes might be more like your right hand fifth finger. They are going to play most of the highest notes.

The trombones are pretty low, so they would often be playing left hand notes. The trumpet on the other hand would be maybe third or fourth finger on the right hand.

Now you might be thinking that for some chords, you might use your right hand third finger to play the note F4, but then on another chord you might use your right hand first finger (thumb) to play A4, two tones higher. That's also what happens in an orchestra. Sometimes "lower" instruments might play a note that is higher than a note played by a "higher" instrument. There's a lot of overlap in the ranges of the instruments, so there can be overlap in what they play.

Now you might be wondering why would there even be multiple instruments in one orchestra or band that can play the same notes. Well they sound different, so you can get different tone colors by having one instrument play some notes versus another instrument. And also you can have different combinations of instruments playing together to get even more sound possibilities. So you might have trumpet, flute, and oboe all play the same melody at the same time and it will have a certain sound. Then later you have trombone and clarinet play the same melody together and it will have a different sound. Then the flute can come in on top and make a third kind of sound. Then they can start playing different parts and you're back to making chords with the different instruments. And which instruments are playing to make the chords also gives the chord its unique sound.

So it's a pretty complicated interplay of different notes and different timbres that make up the full sound that you hear from a band or orchestra. And the possibilities are vast. Perhaps sometimes they are playing all the same notes in different octaves, and other times it's just one lone flute or trombone playing all by itself. Composers mix and match at will to get the sounds they want.

The instrumentation in a march and the use of wood wind instruments can be compared with the registration of an organ.

The different instruments are employed in respect of their pitch level (SATB), as group of a family (brass section/woodwind section, or they are mixed together depending or in purpose of the color of the sound you want to create.

Flutes and Clarinets will support the melody instruments like trumpets and cornets, or have their own part of ornaments and girlandes with the well known trillers (Stars and Stripes).

In the often calmer “trio-section” Clarinets can follow or replace the tenor horns and Euphoniums in the leading voice aswell as in cointerpoint.

In the harmony band the trumpets may be used for the leading voice also for special fanfares effects.

The section of the “bass solo” the lower woodwind will share the tune of the bass instruments and the higher support the accompaniment of the small brass.

There are infinite possibilities of combinations for sounds, colors and effects, playing in harmony or unisono witch will lead to fact that the different sections have more rests and chances to relax.

The simplest way to answer the question is for the OP to actually look at some marching band scores.

There are two "minor" problems with that, of course: one is that there is a lot of garbage published on the web, and the second is that traditionally, marching band scores were often written in a very condensed form and the OP would most likely not be able to figure out "who plays what" correctly from the notation.

A good source that avoids both those problems is https://www.marineband.marines.mil/Audio-Resources/The-Complete-Marches-of-John-Philip-Sousa/ which is publishing the all Sousa's marches in conventional modern full-score format (and each score has a preface explaining any non-obvious details).

That will quickly demonstrate that asking about "trumpet and clarinet playing the same music" is too vague, considering there may be three separate trumpet parts, and as many as 6 clarinet parts, for four different sizes of clarinet!

The site also has audio tracks of each march, recorded live. Some of them are very familiar, of course.

No. Orchestration, whether for string quartet, full symphony orchestra or anything inbetween, is much more interesting than just 'give the higher instruments the tune'.

What band music have you listened to? Try this. Doesn't matter they aren't actually marching.