Questions about alternate characters: What is a good way to use them and should I make one for a different style?

First, what is a good way to use alts(characters, not accounts), and why do people use them?
Second(and more importantly), should I make an alt to use different weapons, or use my main character?

Before release 2013-07-30, each of your "paid" accounts got its own stream of mist energy. Therefore it was a good idea to spend a little bit of real money on each of several accounts, to get several mist streams going. Some old players (such as myself) have alternate knights for this reason. But since release 2013-07-30 there is no longer any practical reason to have alternates.
Some players use alternates because they are into role-playing. Suppose that you think of your knight as a cool, calculating bomber. And your knight's friends expect him to bomb all of the time. But now you want to try gunning, and maybe you also want to try on a spazzy, excitable personality. Then you might make a new knight for that new persona.
Or maybe you like your guild, but you want to see how other guilds work. So you make a new knight and join some other guild.
Other than reasons like that, it's best just to use your main character. That knight has already worked through the rank missions. He (or she) has access to the Hall of Heroes and its recipes. He has extra orbs lying around. He can use old armors while crafting up his new exciting armors. And so forth.

It's basically the "New Game +" of the SK. You have the power to mooch off of your main to farm (say, one with FSC access). I have had 6 characters in my time, and as a F2P player, I only really have the patience and time to do one full 5* set per playthrough. As Bopp has said, I always try something different for each character, such as a Mercurial Mail set for one, pure-swordmaster for another, or pure-gunner.
So really, whenever you're bored of doing the same loadout or tired of T3-grinding, start fresh from the beginning and try a different approach to SK. I always make a new character whenever something big happens, like the Gunner Update or Sprites.
Also RPing is just /sad.

Before when one played on mist, having alts was a good way to circumvent limited playing "time" (100 mist and done). Now, that is not the case, so having alts is a drain on resources as you need to outfit each separately. (Of course, that was the case during "Mist play," but if you wanted to play longer there was no other alternative than to have an alternative character.) I would NOT recommend making an alt as all you need now is the change of gear to play further along according to the missions;you don't need mist to continue playing--just the better gear.
As Bopp said, I think the only practical reason, post-Mist era, for having an alt is perhaps for joining a different guild; or maybe you want a different personality?--such as one character you play sweet, one nasty. lol.
Cheers.

"So really, whenever you're bored of doing the same loadout or tired of T3-grinding, start fresh from the beginning and try a different approach to SK.
Wouldn't it make sense to just keep your fully heated 5 star gear and bring the "new stuff" you would like to get heated? After all, heat doesn't accumulate in fully heated items.

Progress threw all the missions on a new knight, new friends, just starting again.
There is no "easy" way to earn radiants, if that is your goal.