Why is Sword of Body and Mind regarded as the weakest of the Swords?
It's just not very good. Sword of Body and Mind's abilities aren't as powerful individually, and they have very poor synergy with each other.

Sword of Fire and Ice is great for a tempo deck beating down on another tempo deck:


Draw a card: +1 card advantage.
2 damage to target creature: often +1 card advantage. Lots of creatures worth killing have 1 or 2 toughness (e.g. Dark Confidant, Mother of Runes). At worst, it speeds up the clock a little.
Protection from red: negates red-base removal, which is plentiful because Lightning Bolt is SOOOO good.
Protection from blue: very marginal, but means your opponent can't win tempo by bouncing your creature.


Sword of Light and Shadow is a great card for creature-based midrange:


Return a card to your hand: consistently +1 card advantage in a deck with a reasonable number of creatures (16+).
Protection from white and protection from black together in one card: negate almost all of the non-damage-based creature removal in the game. +2 toughness also helps make your creature resistant to red removal.
Gain some life: marginal, but it can help one midrange creature deck beat another in a damage race.


Sword of Feast and Famine is superb in midrange control decks like Cawblade:


Opponent discards: often +1 card advantage. You don't get this if your opponent's already running on empty, but if that's the case you've likely already got them on the ropes.
Untap your lands: this is a massive strategic advantage. Very easy to keep up mana for counterspells or removal while still developing your board.
Protection from black: negates a lot of removal.
Protection from green: a lot of the biggest creatures are green, so it's nice to be able to slip past them. Thanks to the Sword's untap ability, you can easily attack with one dude and then move the Sword to another one so you'll have a pro-green creature to block your opponent's Tarmogoyf. The real value is that it's stapled to protection from black, though: you're giving your creature bonus toughness (to get it out of burn range) and making it immune to black's targeted removal and making it immune to damage from giant green creatures &mdash; sure to give Jund fits.


Sword of War and Peace is a massive clock swinger, useful to any reasonably fast deck:


Damage your opponent a whole bunch if they're hoarding cards: basically makes your creature a very fast clock against control.
Damage your opponent and gain life: creates significant life swings in an aggro/tempo damage race.
Protection from red and protection from white: negates a lot of removal, including cheap removal all-stars like Lightning Bolt and Swords to Plowshares/Path to Exile. Pro white also lets you bypass token swarm defenses like Lingering Souls.


Sword of Body and Mind is, in contrast, a mishmash of abilities that aren't so great by themselves and don't really fit together:


Make a token: technically +1 card advantage, but a Grizzly Bear is kind of a bad card. In most cases this is flat-out inferior to drawing a card, getting a creature back from your yard, or being able to blast something with a bit of damage (since you can remove weenie creatures with actual abilities).
Protection from blue and from green: these are the two colors in the game with the least removal.
Mill: meh. Mill's not always terrible, but here you only mill them when you are inflicting damage. In most cases, that's strictly inferior to any kind of effect that just lets you inflict more damage.