Don't think of them as "better". Think of them as different. Each of the 3 types have different octane ratings. Usually it's 3 of the following depending on your elevation: 85, 87, 89, 91, 93. Octane rating measures the gasoline's resistance to detonation. Detonation is when the fuel in the cylinder ignites before the spark plug actually fires, from the compression of the piston heating the air/fuel mixture to a point where it ignites on it's own. This is bad because it usually means the mixture ignites before the piston has gone all the way up, which is obviously a very bad thing and strains a motor because the hot expanding gasses are trying to push the piston down when the rest of the motor is trying to push it back up. When this happens it sounds like somebody rattling a coffee can full of marbles under your hood.
Higher octane also burns slower, enabling the engine to run a more aggressive timing curve, igniting the spark plugs before the piston has reached the top. More advanced timing = more torque. Too much timing advance = detonation.
When building an engine for high performance, manufacturers will often run more aggressive timing curves and higher compression ratios, two things that help engines make power. Both of these things also tend to make an engine more likely to detonate early for the reason's I've explained above. For these engines, higher octane gasoline may be required to help resist this detonation.
If your engine does NOT require higher octane gasoline, and most don't, then you are hurting yourself by adding plus or premium because your engine was not designed to take advantage of the slower burn you get with higher octane. In many cases your mileage can go down due to an incomplete burn in the cylinder. Combine that with the fact that you paid more for it and it starts looking really stupid.
Most cars made since the late 1980s, and pretty much ALL cars since the mid 90s, have knock sensors. If they require higher octane gasoline and you add regular, the engine is able to detect the knock and will retard the timing for that particular load/RPM to prevent damage to the engine.
There are people out there who buy plus/premium because they think they are being nice to their car/treating it better/whatever, these people are idiots. LOOK IN YOUR OWNER'S MANUAL - it will specify what type of fuel your engine was designed for. This is what you should use. Don't buy into the hype that gasoline makers spew to try and make you think there's something magical or better about premium gasoline.
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(I wrote the PROM tune from scratch for my 1991 Camaro's transplanted 5.7 liter V8, I spent hours experimenting with different spark advance curves, types of gasoline, listening for knock, 3D fuel delivery tables, etc etc.)