The answer is: both. A lot of the pottery we modern folk are most familiar with from the ancient world, like the fine red- and black-figure Athenian wares from the 5th and 4th centuries BCE, were expensive in antiquity and priceless now. Just for the sake of example, we find many items from the set of vessels used by Greek elites to have drinking parties—the accouterment of revelry. Here is a good survey of the types, with pretty pictures. These sets of vessels were very similar to tea sets, with certain vessels for certain functions. Storage of the wine and clean water; vessels to mix the wine and water (only barbarians drank unmixed wine); vessels to pour the mixture; vessels to drink the mixed wine (cups, kylikes, etc). These were what we call fineware: expensive to produce, lavishly decorated, with the highest quality Athenian (or Corinthian) clays.

But less famous but equally present are what we call coarsewares. These are the cooking pots, the shabby tavern cups, the enormous dry stuffs storage vessels (called dolia). These are called "coarse" because their clay fabrics are not as nice, nor generally is their craftsmanship. They are unpainted or painted simply, and their glaze is generally more functional than pretty.

Every archaeological excavation with any sort of domestic or industrial context will unearth tons of the stuff. Many people are annoyed by coarseware, because it gets in their way towards finding the older, prettier stuff they are after. Archaeological layers were deposited in reverse chronological order, meaning that when you start digging in, say, Athens, you first go through the destruction and junk of the World War years, then down into early modern layers, then down into Renaissance and then medieval layers, then down into Late Antique layers and Byzantine layers, then down through centuries of Greco-Roman stuff, then into Hellenistic layers, and then into Classical stuff, then further into elusive Archaic Athenian layers, then into (maybe) some Iron Age Athenian graves, ... on and on, back to the Neolithic and beyond. If you are an archaeologist of the Neolithic, your virgin trench might turn up none of that earlier stuff, or it might turn up 3,000 pieces of Byzantine pottery that must be collected, possibly sorted, and stored in cans for future study (but probably not).

Coarsewares often get a bit of a bad rap, but like all archaeological material, they have data to share. Different types of archaeologists are interested in different things, and some would rather find an abandoned well filled with ancient butcher's trash any day rather than a pile of fineware pottery. Classification of coarseware sherds can be a little more tricky, because (depending on time period and location) they can have a much larger variety of shape, fabric, color, etc. I can look at a palm-sized sherd of Geometric pottery from the 9th-8th century BCE and tell quite a lot: the color and quality of the fabric strongly indicates where it came from (Argos? Corinth? Athens? Lakonia? Euboea?); the painted features, if any, can help me pinpoint the date (is it Early Geometric? Late?); its shape, even when broken, can tell me what type of vessel (is this part of the handle of a skyphos? is this the lip of an SOS amphora?). But a piece of coarseware from the same trench can be very much harder to figure out, just simply because there are less clues (like painted decoration) and I am very much less familiar with the range of shapes, fabrics, etc.