The second one at the end of the month is the proverbial blue moon of the phrase "Once in a blue moon." They occur somewhat randomly every few years, to give context for the actual meaning of that phrase.
To approach a more accurate "actual meaning" of the phrase, it wasn't originally the second full moon in a month, but instead when a season contained a month with two full moons then that meant that the third full moon of the season was the blue moon.
Quoting the Wikipedia entry for blue moon:
"The author of a March 1946 article in Sky & Telescope attempted to decipher the traditional practice of the editors of the Maine Farmers' Almanac by examining old issues of the almanac. Without enough almanacs to see the correct pattern, he conjectured the wrong rule for 'blue moons', which led to the modern colloquial misunderstanding that a blue moon is a second full moon in a single solar calendar month, with no link to the order it occurs in a season."
I can see why he'd make the wrong conjecture. The actual rule as you state it is bizarre.
There are almost 12.4 full moons per year; that's more than three per season. Every season will have at least three full moons, so the fact that a season does have three is completely unworthy of notice.
The fact that a month has two full moons is worthy of notice, since calendar months are just barely longer than lunar months. But when you do have an unusual month... why would that trigger a special designation on an unrelated full moon that isn't out of the ordinary in any way?
Wikipedia's description does make more sense; notice is not taken of whether a month has two full moons. (Though the condition is necessary and sufficient.) Instead, the "blue moon" is called that by virtue of being the extra full moon that occurs in a full year when that year has 13 full moons instead of 12. ("Folklore assigned a unique name to every full moon of the year.") This had to be either the second or third full moon in the season containing the extra moon (since the first / fourth full moons of that season signal its beginning / end).
Not co-incidentally, the proper sidereal solar months (~ Indian solar calendar) also have two new-moons.
This is considered a special enough occasion to warrant another month on the lunar calendar. I half-kid ofc. (in India, everything is 'divinized', and filled with wonder); this is actually a synchronization mechanism for the Indian luni-solar.
I've also read (but can't find the links for numbers) that the brightness of a full moon compared to just a couple of days before and after is on orders of magnitude brighter.
It might be that a supermoon is 150% brighter than 2 days before, for example.
This link has a nice picture that shows the full moon at perigee (supermoon) vs the full moon at apogee (micromoon). The average moon is not pictured, I guess it would be the midpoint.
Full moons are terrible for stargazing. Good bye night vision. A full moon is the least exciting moon to look at it with binoculars (apart from a new moon, I suppose). Look at it in any other part of the cycle and you can see 3D detail on the surface. Much more exciting.
People think of "Tuesday night" as the night that falls at the end of Tuesday, however the majority of it from an astronomy perspective tends to fall after midnight (particularly with daylight savings time), hence it's referred to as the 2nd.
