As asked in the question, also a bonus question; how does the gauge of a guitar affect the tension of the strings?
Strings gauged at 0.011 and 0.012 differ in diameter by 0.001 inches.
Strings for guitar are usually sold in sets and named after the lightest string in the set, so the implication is that, in a set of "elevens", all the strings will be thinner than the corresponding strings in a set of "twelves". Different manufacturers have different combinations with names such as slinky, light, medium, heavy, etc. The gauges of the other strings will differ from one brand to another.
Gauge and tension have no relationship until you bring frequency into the picture. Assuming scale length and material (nickel-plated steel, for example) are constant, a string of heavier gauge at the same tension as a string of lighter gauge will produce a lower frequency. So, all other things being equal, if you use a set of heavier strings, you will put more tension on your guitar to bring it to standard tuning.
You may find this article interesting: http://www.daddario.com/DAstringtensionguide.Page
In 'guitarist' rather than 'physics' terms & assuming the same guitar/string type/tuning in each case...
A set of 11s is easier to play than a set of 12s, but not by much. The extremes of 8s up to 14s would really show you the difference in playability & how easy it is to bend a note. By the time you're up to 14s, you are really in 'Do I have the strength to hold this chord down?' territory.
8s or 9s would lend themselves to 'fast-fingers & heavy styles, as you would need a high gain amp to make them sound any good at all.
By the time you're getting up to 11s, the top end is going to start sounding good with less gain/distortion - & as a personal preference I use 'Skinny Top Heavy Bottom' which start at 10, but then increasingly over-emphasises the gauges of the lower strings to keep them tight & crisp too, the low E being a 52 rather than the standard 46.
Ultimately, no-one can tell you what gauge to use, you have to try them & see - but one important thing to bear in mind is that if you swap string gauge, increasing or reducing overall tension on the neck, the neck itself will need adjustment to keep it straight; not a job for a beginner.
I can only guess that you mean .011 and .012 gauge strings, as often found as the top E on guitars.The numbers refer to the diameter in thousandths of an inch - 11 thou and 12 thou. For the same speaking length, and same material, .012 string will be tighter, making it slightly harder to press down or bend. The sound will be a little richer, as it's moving a bit more air, or presenting a bigger magnetic field in the case of guitars with pick ups.An estimate would be about 10% tighter.