In the UK at least "Premium" 95 RON and "Unleaded" 95 RON will exhibit very little difference - there might be some differentiation in terms of cleaning additives etc added to those listed as "Premium" but there's no standardised or official definition of the term.
As you mention "Super" unleaded fuels have a higher Research Octane Number (RON) value (97 for most, 98 for BP Ultimate, 99 for V-Power/Tesco Momentum, 102 for BP Ultimate 102). In laymans terms the RON value for a fuel indicates how much compression it can withstand before detonating. In a normal combustion cycle the air/fuel mixture is compressed by the piston to heat it before being ignited by the spark, if the fuel detonates spontaneously under the compression this (generally called "knocking" or "pinging") is a Bad Thing and can cause significant damage to the engine if it occurs repeatedly.
Modern engines have a so-called "knock" sensor that detects this and alters the ignition timing accordingly to prevent it at the cost of some performance. Conversely a higher RON fuel allows the engine to run at it's intended performance levels - but and this is the key thing, this can only go as far as the engine map allows. Putting 99 RON fuel in a car that has no ability to take advantage of that will make zero difference and conversely putting 95 RON fuel in a car that is only equipped to adapt as far down as say 97 RON (such as most imported JDM Imprezas) will result in frequent knocking and a one-way ticket to the scapheap for the engine.
Just to further add to the confusion potential you'll see seemingly lower octane ratings listed in the US - this is because the Octane value used for US fuels is measured differently, using the Anti-Knock Index (AKI) - being the average between the Research Octane Number and the Motor Octane Number (MON). MON is also an indicator of knock-resistance but is calculated using a different test and there's no direct relationship between the two.