In this work, we present the effect of silica’s specific surface area (180 m2·g−1 and 380 m2·g−1, respectively) on the crosslinking of styrene–butadiene rubber (SBR) composites, as well as their crosslink density and functional properties, such as thermal stability, damping behavior, resistance to thermo-oxidative aging, and tensile properties. Ionic liquids (ILs) with a bromide anion and different cations, i.e., 1-butyl-3-methylimidazolium (Bmi), 1-butyl-3-methylpyrrolidinium (Bmpyr), and 1-butyl-3-methylpiperidinium (Bmpip), were used to enhance the cure characteristics of SBR compounds and the functional properties of SBR vulcanizates. It was proven that apart from the silica’s specific surface area, the filler–polymer and filler–filler physical interactions have a significant impact on the vulcanization kinetics of silica-filled SBR composites. Additionally, the performed studies have shown that ILs positively affected the dispersion of silica’s particles and reduced their ability to form agglomerates in the elastomer matrix, which enhanced the functional properties of the SBR vulcanizates.