Basically, not that you should ever put two bettas in together or anything
and yes, the first picture is a picture of a halfmoon betta, while the second one isa veitail. There are different types of betta splendens and they are classified by teh shape of their tail.
There are crowntails, veiltails (the most common ones), halfmoons, doubletails, combtials.
Veiltails are the most common ones and usually the ones you see at petstores, they're no longer really considered as "good" betta (though i think they are still beatiful none the less).
Crowntails look like this:
doubtails literally have two tails, like this:
halfmoons are the first betta that you have on your list, they are very very pretty becuase their tails spread 180 degrees.
combtails are a mix of crowntails and veiltails, so the crowns are not as distinct.
HTH
p.s. there's no real reason to "why" this happens, its' like poeple having curly hair or straight hair, just genetics