The capacitance formed between copper planes in different board layers is typically much smaller than the capacitance from discrete capacitors. but it has a very low parasitic series inductance and therefore is the most important capacitance for frequencies above a few 100 MHz. This frequency range is important for decoupling fast ICs or for handling ESD.
The capacitance between a trace and the plane underneath is important for transmission line impedance. A thinner dielectric reduces the characteristic impedance at the same dimensions. therefore trace dimensions can be smaller for the same characteristic impedance, which facilitates dense high-speed routing.
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