In ceramics, porcelains lack plasticity and fire to high density with white, glassy surface. Stonewares are plastic and fire cream to brown and lower density, Porcelaineous stoneware are between these two.
Conceptually, a high temperature white porcelain could have recipe of 45:30:25 kaolin:feldspar:silica (typically high quality kaolin is employed and a bentonite addition is also needed to augment plasticity). It would fire dense, with zero porosity. If the kaolin is low in titanium and iron it could also be translucent. If the kaolin and feldspar and low in iron it could fire super white.
A stoneware could be something like 60:20:20 ball clay:feldspar:silica. There is more clay, and that clay is much more plastic than kaolin. There is less feldspar so it vitrifies less, perhaps having a porosity of 1-2%. It is thus less prone to warping in the kiln. And has zero fired translucency. There is less silica because the ball clay contains a percentage naturally. Often significant iron content in the ball clay can be tolerated, producing a body that fires tan or even brown. An important difference is that the stoneware has much less expensive materials. And it is much easier to form and dry. And easier to fit glazes to. It can fire anywhere from bone to brown, but for many types of ware that are completely glazed, the body color is not a big issue. Or, the iron in the body bleeds into the glaze providing an attractive aesthetic. Typically, the raw materials are industrially processed at 200 mesh particle size, however some producers can tolerate more coarsely ground clay. When this is the case, fired density and strength normally suffer somewhat.
Porcelaineous stoneware is somewhere between these two, I believe the term is best understood from the context of data, produced by experience in testing many types of bodies, measuring their physical properties and understanding these in terms of their recipes and firing circumstances. This perhaps a somewhat 'fuzzy' concept is betrayed by the variety of ways it can be spelled. But we can generalize that porcelaineous stoneware would typically always be using 200 mesh materials. "Porcelaineous" could mean that it needs to fire whiter. This could be accomplished by using lower iron ball clays (or by a mix of ball clay and kaolin). "Porcelaineous" could mean that it needs to be more vitreous than the stoneware, having a denser and more glassy surface or have more durability, hardness or strength. This could be accomplished by raising the feldspar to 25 or 30 (at the expense of the clay). "Porcelaineous" could also mean that a smoother plastic material is needed to create finer detail or produce a smoother fired surface. "Porcelaineous" could also mean that a producer can afford to spend some extra money on the body to achieve a combination of smoothness, whiter firing and better durability (at the cost of some workability).
All sorts of other materials are used in porcelains and stoneware, but this is a conceptual explanation of the differences. The whole topic is quite subjective. Many available clays are not really ball clays or kaolins, they are natural mixes of multiple clay minerals as well as feldspar and quartz. These are commonly employed in stonewares but much less likely to be used in porcelaineous stonewares (unless very finely ground). Fireclays are also often used in stonewares, but because they have coarser particle sizes, would not be employed in porcelaineous stonewares. Another factor to consider in terminology. People often refer to "vitreous stoneware", on doing that there may or may not be referring to porcelaineous stoneware (the term vitrification is not really a state but a process).
Left: 65% #6Tile kaolin and 35% nepheline syenite. It's white but crazes the glaze and has 1% fired porosity (measured in the SHAB test). Thus it does not have porcelain density. Plasticity is very good. Right: 65% M23 Old Hickory ball clay and 35% nepheline syenite. The glaze fits, the body has zero porosity (very dense) and plasticity is fantastic! The body on the left needs a 20% silica addition (to stop crazing) and 5% more nepheline (to reduce porosity to porcelain levels). But the remaining 40% kaolin will not be nearly enough for a workable plasticity (so bentonite will be needed). The body on the right does not need fixing because ball clay is easier to flux with feldspar and it contains its own natural silica.
Ball clays are normally refractory, none of these are vitrified to any extent. The cone 10R bar is yellow because it is stained by the soluble salts present in the material. These are very typical of what most ball clays look like.
It is amazing now similar different ball clays are. Clearly, soluble salts are an issue with all of them (the brownish scum). These bars are much cleaner on the backsides (since the solubles were left on the surface on the fronts during drying). The drying shrinkages, plasticities and fired maturities are also all remarkably similar.