Before explaining why this type of Roman concrete is different, let me explain what typical concrete is. Concrete is composed of 4 primary materials:
- Fine aggregate (sand)
- Coarse aggregate (gravel)
In typical concrete, the cement is Portland cement therefore it is typically referred to as Portland Cement Concrete (PCC) to distinguish it from Asphalt Concrete, which also has fine and coarse aggregate, but uses Asphalt for the binder instead of Portland cement.
There are several varieties of Portland cement that are given roman numbers I to V and are used for various purposes
Type I - General purpose
Type II - Moderate sulfate resistance
Type III - High early strength
Type IV - Low heat of hydration
Type V - High sulfate resistant (to avoid a damaging chemical reaction ASR)
Portland cement is not the only binder that can be used in concrete, it is just very common, available (it is not a natural material rather it is manufactured), strong, and has reasonably good durability.
Other materials that react with water to form cementicious binders are called pozzolans. These can be various natural materials or often times are byproducts of other manufacturing processes. For example fly ash, silica fume, coal ash, slag, etc.
These alternative pozzolans are often substituted in for Portland cement in various quantities. Typically it is not a full replacement, but rather a partial replacement. The exact ratio of Portland cement to pozzolan replacement vary by the mix design and pozzolan type. You can read more about them here.
Cement and pozzolans react with water to make various chemical compounds that are primarily composed of calcium-aluminum-silicate-hydrate (CASH). The exact compounds that are formed depend on the type of cement used. This is the key part of why they believe the Roman concrete was so strong, and is the key focus of the article you linked. From the article
The concrete foundations, walls, and vaulted ceilings are composed of decimeter-sized volcanic tuff and brick coarse aggregate (caementa) bound by volcanic ash–lime mortar.
So they just mentioned that they used volcanic tuff and brick as the aggregate, and they used volcanic ash mixed with lime as the pozzolan.
Remember that we want the cementitious material to form CASH compounds to be an effective binder. Volcanic ash is primarily $SiO_2$ or $AlO$ compounds, and lime is primarily $CaO$. When you mix those with water you have essentially
$$CaO + AlO + SiO_2 + H_2O \to CASH$$
Note that the Romans did not have Portland cement at the time, so this was complete replacement as far as the pozzolans are concerned. The authors found that a particular compound is formed during the reaction called strätlingite and note that
The strätlingite crystals in Imperial Roman mortar resemble microfibers that are added to the cement paste of present-day mortars and concretes to produce toughening—except that they crystallized in situ and reinforce interfacial zones, the most vulnerable component of the mortar fabric.
The authors emphasize the significance of these crystals
Three principal differences exist between authigenic strätlingite crystals in Roman mortar and microfiber additives in present-day cement pastes.
First, radial spherulites of strätlingite grow preferentially in scoria interfacial zones, whereas microfiber additives remain in the cement paste and do not reinforce aggregate interfacial zones.
Second, strätlingite is resistant to corrosion, relative to glass and steel fibers in cement matrices.
Third, authigenic crystallization of dense strätlingite intergrowths occurs in the complex accretionary perimetral zones of scoriae, groundmass of scoriae, and cementitious matrix (Fig. 3F) long after portlandite was consumed at about 90 d of hydration.