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Voyager "Year of Hell":

JANEWAY: It's beautiful.

CHAKOTAY: Nineteenth century, mechanical movement. It's a replica of
the chronometer worn by Captain Cray of the British Navy, His ship was
hit by a typhoon in the Pacific. Everyone back in England thought they
were killed, but eight months later Cray sailed his ship into London
harbour. There wasn't much left of it, a few planks, half a sail, but
he got his crew home.

JANEWAY: I appreciate the sentiment, but I can't keep this. Recycle
it. We can't afford to waste energy on nonessentials.

CHAKOTAY: Kathryn, I replicated this months ago. I've been saving it.
I wanted you to have it.

JANEWAY: That watch represents a meal, a hypospray, or a pair of
boots. It could mean the difference between life and death one day.

Wouldn't' it take energy to recycle something? If all they're looking for is raw materials, meaning matter, there's tons of wreckage all over the place. If all Janeway needed was metal which is what the watch is made of it was all around here. I was confused by this concept. Is replicated material special in some way once it's completed?

How does "recycling" this watch in some way benefit them?

The replicator is a fancy 3D printer. It has a stock of standard raw material (base atoms) that it changes and arranges on demand. Nothing special about its products. But it has consumables, energy and raw material.

The use of the word energy here has a few meanings. First it could have been she misspoke, having meant to say material instead of energy. The meaning would stay the same. The second would be energy as in time and effort. Janeway thinks it's frivolous compared to the current needs. The third does mean energy. While the warp reactor can provide enough energy in normal circumstances, it does cost dilithium crystal stress. Recycling it might not help much, but Janeway would never let any crew member, let alone a senior officer, break an important rule during that time. Survival comes first. She's setting an example.

Since Voyager at this time is rationing replicator usage because they can't replenish supplies or undergo routine maintenance. Basically the only reason people try Neelix cooking.

Non-canon (as far as I know) but the only reasonable explanation is that replicated matter is not the same as ordinary matter at the quantum level. A replicator can "recycle" replicated matter, turning it back into energy - but cannot do the same to ordinary matter.

(If a replicator could turn ordinary matter directly into energy, and back into a different form of matter, there would have been no reason for rationing.)

Compare this answer.

Not correct, but good guesses.

According to the Star Trek the Next Generation Technical Manual, the replicators work off of "raw replicator matter" which is essentially simple molecules that can be reformed into anything desired. Using normal matter, such as stuff from the recycling bay, cost more energy to use the replicators.

Using this recorded fact as a basis, I can only presume that a replicated meal, even non-organic "raw matter" to make a watch, was at its core this same string of simple molecules that can easily be reversed into its raw form.

Capt. J. Harker, USS Twilight

The replicator turns energy into mater and mater into energy, but energy is lost in the process. It actually makes little sense to convert it back into energy for making something else later. Too much energy is lost in the process.