NOT to "Pick Nits" here, but at the same inflation pressure ANY size tire will have the SAME "rubber on the ground".
Changing tire width &/or profile changes the SHAPE of that "footprint" leading to advantages or disadvantages in different circumstances.
Wider tires give a shorter, wider footprint - an advantage at high speeds because of less "contact time" heating up the tread - a disadvantage for Snow Tires because the added width causes increased flotation (wider also needs larger rain groves to counteract hydroplaning for the same reason)
Lower profile, commonly used to keep the same overall diameter when adding width, gives quicker steering response due to the reduced sidewall height vs. width. This gives the "Sportier" feel usually attributed to wider tires.
Wider rims to match the wider tires add greatly to the quicker steering response from lessened sidewall deflection. Extreme version is to use wheels that are wider than the tire's tread (stretched). Quite common in racing applications, and a dramatic difference even if the wheel is only slightly wider than the tread width (for example a 5" tread width on a 6" wide rim).
Plenty more "fine points" to the subject, but that gives the general idea...