In one episode of ST:TNG, a transporter incident creates two copies of Commander Riker. Why doesn't Starfleet (or anyone else, like the Romulans) use this to create copies of priceless historical artifacts or clones of particularly skilled or important individuals?
That was a freak accident that Federation physicists didn't anticipate, could barely understand and couldn't come close to recreating. That episode was all about coming to terms with that novel situation.
Replication is a well-established Federation technology; it has limited fidelity -- replicated objects are riddled with errors at the atomic level -- so it's good enough for food, bulk materials or technology designed with replication in mind (i.e. fault-tolerant) but a replicated organism arrives dead. (This just happens to coincide with the ancient superstition that a living thing is fundamentally different from a non-living one.) I'm sure that replicated artifacts are as common in the federation as recorded music is in ours.
If real duplication technology existed in the Federation, the Federation would have adapted to it, and the result would not have borne much resemblance to the Star Trek we know.
Replicators do use transporter technology, according to MA, but there are limitations inherent in each. OOU, these differences are meant to prevent either technology becoming all-powerful, so scriptwriters don't try to use it to solve everything. In-universe, the difference is primarily based on the vast difference in the resolution of the "patterns" employed in either case.
In matter transportation, the person's physical pattern is analyzed, and then their matter is disassembled and transmitted as energy to the destination where it is reassembled. It's hinted at many times through the canon that the pattern data is stored in a spacious but highly volatile "buffer" allowing short-term WORO (write once, read once) storage during the transportation process; "saving" a person's pattern long-term using traditional WMRM (write many, read many) data storage requires a lot of memory ("billions of kiloquads"). In an episode of DS9, this was done, but the entire non-essential data storage capacity of the entire station only held something like 5 full patterns.
Replication is similar, but instead of starting and finishing with the same thing, you start with bulk matter and rearrange it into whatever you wanted based on a stored pattern. The limitation is pattern resolution; there simply isn't enough space in the ship's computer to store all the pattern data required to perfectly replicate everything the replicator can produce, so the "resolution" of the replicator's stored patterns are instead reduced to something "close enough" to produce something useable; food, clothing, simple objects and machines. The maximum resolution of the replicator's actual molecular assembly matrix may also be reduced, as the extra resolution inherent in a transporter would be superfluous given the previous point, that nothing stored as a replicator program would require this level of detail to reproduce.
This resolution is still fine enough that, to most senses, the objects produced are indistinguishable from "the real thing". Musical instruments, for instance, are a not-uncommon request of a replicator, and the end result incorporates both the nuances of design and the musical properties inherent in the structure of the materials. However, it's virtually always possible to tell the difference, even if the only way to do so is scanning at the molecular level. In addition, because of this inherent reduction in resolution, things that do require the highest level of detail, such as replicating human organs or entire living things, are simply not possible with the traditional replicator, because there's not enough information available about said thing to accurately reproduce it.
The actual materials may also be in short supply. Latinum, for instance, is composed of exotic matter sparsely available in certain nebulae, which can be harvested and suspended within gold (the value of the gold itself is stated to be relatively low). Latinum is the actual matter and cannot be broken down into anything smaller or simpler; therefore, to replicate latinum, you must have latinum, and so you're really just transporting it from the storage units into the replicator bay. Antimatter, for similar reasons, cannot be replicated, though there are other ways to produce it artificially.
Industrial and medical-grade replicators are both reported to exist in canon, but not aboard ship. The apparent tradeoff is a bigger replicator that can produce very finely-detailed complex objects, but the information required for the pattern is higher and so the replicator doesn't have as large a "library" (it may, in fact, only have enough memory behind it to produce one thing, and must be "retooled" to change what it makes). The energy demands may also be higher.
Lastly, as was already mentioned, there are ethical taboos against genetic engineering and cloning in the Federation as a result of Earth's Eugenics Wars in the late 20th Century, and this taboo holds in most (but not all) other factions in known space. So, even if someone did have access to a replicator with the information capacity and resolution to reproduce a living thing, if he actually did so he would likely not have that access for long after his attempts were discovered.
The short answer is that because it was an accident, the repeatability of such a process is doubtful. Something that was caused by a malfunction may not be a process that can be safely used on a regular basis.
We have also seen people die in transporter accidents several times in Star Trek films and television shows, so it is probably unwise to risk a person or a valuable artifact by transporting it unnecessarily.
This brings up an interesting issue. One of my major gripes with Star Trek is the overuse of the transporter technology. Only Drs. McCoy and Pulaski seem to share the view, which I personally feel most of humanity would subscribe to, that the transporter is too dangerous for use by people except in the direst of circumstances. There was a discussion just last week on Australian radio about future uses of transporter technology, actually, and physicists seem to universally agree that transporters would be useful for cargo, not people. But I digress.
As Stan said, there are potential ethical and legal ramifications behind creating exact duplicates of both people and important cultural artefacts, not to mention the obvious copyright issues with newer objects. It is also entirely possible for the Federation to use transporter technology to scan an item, but not follow through with actual transport. That is how items are programmed into the replicators, and that technology is itself prone to frequent errors. It is entirely possible that the Federation, along with other polities, do use such a process to safeguard important items. We also know for a fact that the Klingons used a similar process on their legendary first Emperor, Kahless.
Lastly, there is a taboo against genetic engineering and cloning in the Federation and Klingon Empire. While the creation of transporter duplicates is not encompassed by this ban on cloning, it would seem probable that the Federation frowns on the concept, and if such a process became viable the Federation would be very likely to forbid it.
The teleporter is a widely underused instrument. The machine's abilities are immense and the lack of use they get has baffled me frequently.
The cloning of Will Ricker was an accident. I believe (but cannot support with sources) that cloning is illegal in the Federation, also because of the whole "Eugenics Wars" thingy. The ethics of cloning are very murky and it might simply be considered undesirable.
The information about artifacts and their molecular structure are no doubt saved perfectly somewhere. Yet the beauty of an artifact is that it's original. Man-made diamonds are "fake diamonds" and only natural ones are considered real. That's just life.
Besides, what's the point of having artifacts around anyway? Even nowadays we see people selling antiques and old silverware (and all things of the kind) cheap because they simply have no purpose in their lives.
So: because cloning is unethical/illegal and there's no such thing as a priceless duplicate of an artifact.
The answer to why they don't do it is a moral one that the Riker episode referenced in the question delves into throughly, to say nothing of the fact that the episode-referenced example is a freak accident of transporter technology, but there's also a technical limitation behind why they don't 'back up' important individuals with transporter beams.
In Our Man Bashir, Sisko, O'Brian, Kira, Worf, and Dax are all caught in a transporter accident, and their bodies fail to materialize on the transporter platform. With their patterns in the buffer, the computer is instructed to store the information anywhere it can find. This instruction causes the entire space station to lose almost all power, as it becomes a storage space for their memory - with the information on their physical appearance being stored in the Holosuite (which is noted in the episode to be magnitudes more complex than any other system on the station).
Unfortunately, the explosion comes during the transport, and the crew members' patterns are stuck in the transporter buffer. Given the immense amount of space required to store neural information and the fact that the buffer will soon lose coherence and the signatures with it, Eddington orders the computer to wipe all memory necessary in order to save the patterns
Eddington announces that the neural patterns of all five officers were stored in the rest of the computer memory. They decide to use the Defiant to reassemble the neural and physical patterns of the five victims.
In short, to store the information of just five people required the resources of an entire space station, plus however much data storage it takes to run the Holodeck (with safties all turned off, of course), and restoring them from memory required the station be hooked up directly to the Defiant.
And this too was largely an accident, requiring the instructions for the station be just vague enough to work. Technologically, it's just beyond the realm of unfeasable.
Items are priceless because they are rare.
If you make lots of copies they would no longer be rare, and therefore not be priceless, but worthless.
Similarly important people are valued because they are unusual and unique, they stand out from the crowd. But if you cloned a lot of them, they would no longer be unique.
You may place great worth on an object, someone else considers junk. Value is very subjective.
From a more Doylist perspective, duplication of people should be impossible (Thomas Riker being a weird exception) because Star Trek has always believed in souls, though it usually doesn't call them that. That's why there are so many episodes about people's energy being transfered around, put in computers, put in other people, floating around in energy fields... Transporting might scramble your physical body, but your energy-based soul remains intact throughout. Star Trek generally holds that making souls is not easy, and definitely not something that should be undertaken lightly. So you can't duplicate people.
As for duplicating things, I think the argument would be that if you make a copy of the Mona Lisa, the copy isn't the same. It's not the atoms that Leonardo arranged, it has no connection to the living genius that the original does. I'd argue that since it has no soul, transporting it would also destroy the original, but it's not clear to me if the characters in Trek support that.