Though they are related molecules, they are technically different. You can see Cyanuric Acid here where you can see the main ring of carbon (not shown as "C" because each intersection of lines implies a Carbon) and Nitrogen. Now look at Dichlor which is Sodium Isocyanurate as seen here and notice that though there is the same ring, that now there are two chlorine where hydrogen used to be and more importantly there is a sodium where one hydrogen used to be.
The faster dissolving of Dichlor is mostly because it is already a salt with an extremely weak bond between the sodium and the main ring. Generally speaking, when there is a very weak bond due to high polarity (the Nitrogen is very negative while the sodium is very positive), the substance will dissolve readily as two charged substances are formed and water likes charged substances (ions) because water itself is very polar (the oxygen is very negative while the hydrogen is very positive).
Cyanuric Acid, on the other hand, does have some weak bonds between the oxygen and the hydrogen, but this isn't as readily given up as with the sodium and nitrogen.
You might say, well gee, why isn't there a form of cyanuric acid where one of the hydrogen is replaced with a sodium. Then it should behave more like quick-dissolving Dichlor but won't have chlorine. It turns out there is such a product called Instant Pool Water Conditioner from Natural Chemistry here. Though this is a quick dissolving CYA, it was difficult to manufacture into a stable slurry. I had an E-mail conversation with the chemist who created this product. It took many years to figure out the manufacturing process but the end result is what you would expect. It's just not easy sometimes to create what you want -- creating the monosodium salt of cyanuric acid was not easy.