How do the special abilities of each character translate to game play?

Wilson

Base Sanity: 200
Base Health: 150
Base Hunger: 150

He grows a beard completely on his own. The beard provides the much valuable Beard Hair, as well as 10 sanity on shaving and insulation to prevent freezing during Winter.

Specifically, that translates to the beard growing in four stages:

No beard

Yields no beard hair
Grants no insulation

Short Beard

Grows from no beard after 5 days
Yields 1 Beard Hair on shave
Insulates for 15 seconds

Long Beard

Grows from Short Beard after 4 more (total of 9) days
Yields 3 Beard Hair on shave
Insulates for 45 seconds

Magnificent Beard

Grows from Long Beard after 6 more (total of 17) days
Yields 9 Beard Hair on shave.
Insulates for 135 seconds

Willow

Base Sanity: 120
Base Health: 150
Base Hunger: 150

She has a pretty strong obsession with fire, which translates to immunity, as well as her lighter. The lighter grants an infinite +0.6/min sanity boost while in use and can be used to set fire to ALL THE THINGS.

She will, however, (at low sanity levels) also randomly set fire to things.

Wolfgang

Base Sanity: 200
Base Health: 200
Base Hunger: 300

He gets a strength bonus depending on how well-fed he is. This translates to several appearance and stat changes, depending on his hunger stat - which decreases 1.5x as fast.

Between 0 and 100 hunger

Health between 150 and 200
Deals between 0.5x and 0.75x damage

Between 100 and 225 hunger

Health is at 200
Deals standard damage

Between 225 and 300 hunger

Health is between 200 to 300
Deals between 2.25x and 2.0x damage

He also has a Sanity penalty, however, whereby he loses 10% more Sanity from monsters and in the dark.

Wendy

Base Sanity: 200
Base Health: 150
Base Hunger: 150

Wendy carries with her Abigail's Flower, which can be used to summon Wendy's dead twin sister Abigail.

Abigail's Flower is used when killing any environment mob (including Wendy herself) near the flower after placing it on the ground. This summons the Abigail and drains Wendy's Sanity by 50 points.

Abigail has a Health of 600, deals damage depending on the time of day (10 at day, 20 at dusk, 40 at night) and will mercilessly murder anything that attacks or is attacked by Wendy.

WX-78

Base Sanity: 100
Base Health: 100
Base Hunger: 100

This funny murderous robot has two special powers. For one, it acts like a Lightning Rod, in that it draws in lightning strikes. When struck, WX-78 loses 33 Sanity, is lit up like a torch, gets healed and sped up, entering SYSTEM OVERLOAD. SYSTEM OVERLOAD lasts about one day and has the side effect of being able to set fire to things by using the glowing metal instead of conventional fire.

The second power is the ability to upgrade the robot. Using Gears, WX-78 refills 60 Health, 50 Sanity, and 75 Hunger.

On top of that, each upgrade adds 20 max Health, ~6.66 (alternating 7/7/6) max Hunger, and ~13.33 (alternating 14/13/13) max Sanity. As WX-78 can only use 15 gears for this effect, his stats increase to the maximum of 400 Health, 200 Hunger, and 300 Sanity.

Wickerbottom

Base Sanity: 250
Base Health: 150
Base Hunger: 150

She has a massive distaste for spoiled or raw food. While green-marked food restores the normal values of Health, Sanity and Hunger, any yellow-marked food will only restore 1/3 (instead of the usual 2/3) of its Hunger base value, 1/4 (instead of 1/3) of its Health base value and no sanity. Red-marked food will only restore 1/6 (instead of 1/2) of its Hunger value, no Health and decreases Sanity by 10.

Wickerbottom can also craft Books that grant varying effects and does not require a Science Machine before building prototypes for the crafting recipes that usually need one. This propagates further: Alchemy Engine recipes only need a Science Machine instead.

Birds of the World

Is used to call 20 to 30 birds
Has 3 uses
Cost per use: 50 Sanity
Recipe: 2x Papyrus + Egg

Applied Horticulture

Is used to instantly grow any (fertilized) crops, plants or trees during Summer
Has 5 uses at a range of 30
Cost per use: 33 Sanity
Recipe: 2x Papyrus + Seeds + Manure

Sleepytime Stories

Is used to putting animals to sleep
Has 5 uses at a range of 30
Cost per use: 33 Sanity
Recipe: 2x Papyrus + 2x Nightmare Fuel
Requires a Prestihatitator

On Tentacles

Is used to create 3 tentacles
Has 5 uses at a range between 3 and 8
Cost per use: 50 Sanity
Recipe: 2x Papyrus + Tentacle Spots
Requires an Alchemy Engine

The End is Nigh

Is used to create massive lightning strikes (16 to the count)
Has 6 uses at a range between 3 and 15
Cost per use: 33 Sanity
Recipe: 2x Papyrus + Red Gem
Requires a Shadow Manipulator

Woodie

Base Sanity: 200
Base Health: 150
Base Hunger: 150

Woodie has his precious Axe named Lucy. Lucy deals 13.6 damage, has infinite durability and cuts through wood like a knife through butter.

On top of that, Woodie is cursed. He hides a terrible secret. He will turn into a Werebeaver after a while. This is triggered either upon chopping down too many trees in too short a time, or at Full Moon.

While a Werebeaver, Woodie focuses almost entirely on wood, gaining a "Log Meter" in place of the usual UI. This log meter determines how long he will stay a Werebeaver and reduces at 2 points per 2.5 seconds.

In order to increase the Log Meter, Woodie has to consume wood. All the wood. This increases the Log Meter by a set amount of points per wood item:

Living Log: 50 points
Boards: 15 points
Logs: 10 points
Saplings: 10 points
Berry Bush: 10 points
Spiky Bush: 10 points
Grass Tuft: 10 points
Twigs: 5 points
Pine Cones: 2 points
Cut Grass: 1 point

On top of that, the Werebeaver deals 51 damage like a Tentacle Spike and absorbs 80% of all inbound damage into his pelt. While Woodie is a Werebeaver, his log meter is his health; being hit will drain his log meter.

Careful, though: The Werebeaver cannot carry anything. He will turn and leave everything on the ground - including the precious Lucy, who will then give a heartwarming welcome once Woodie returns.

Woodie will, once the log meter hits 0, wake up the next day - with 50 of all stats - wherever he collapsed when his log meter expired; this can be lethal if he was in combat when his log meter ran out. His belongings will still be wherever he dropped them when he first changed, which can be problematic in the winter.

Wes

Base Sanity: 150
Base Health: 113
Base Hunger: 113

Being the pantomime that he is, Wes carries with him a Pile o' Balloons, which can be used to spawn, well, balloons, at the cost of 5 Sanity per balloon. They act as weak decoys, distracting mobs from the player and will cause 5 damage when popped.

Wes' hunger will drain at 1.25x speed, and he will only deal 0.75x damage.

Maxwell

Base Sanity: 200
Base Health: 75
Base Hunger: 150

Why, why, here comes the big guy. Maxwell, being the overlord of the realm, has with him the Codex Umbra, a dark magic book. At the cost of 15 Health and 55 off the maximum Sanity as well as 2 Nightmare Fuel, the Codex Umbra can be used to spawn a Shadow Puppet, a Shadow Creature to do the dirty work for Maxwell.

The Shadow Puppet has a Health of 75, deals 40 damage and will return the 55 max Sanity to Maxwell on its death. It will help Maxwell murder puny enemies and innocent, but totally valuable and destroyable trees and rocks.

Wolfgang is strong, but only when he's well fed. Food becomes very important. In his mighty form, hunger will also drop faster. This means you'll be eating and eating a lot, to enjoy the characters perks. While other characters have no difference between 300 food or 1 food (except for "time to next meal"), Wolfgang does. You'll be snacking constantly.

WX78 is a very special story. Yes, he's upgradable, but he starts out weaker than most. You'll be looking for the chessboard or ruins pretty fast, trying to find some helpers to slay your foes (yay rock lobsters!). Once upgraded, his stats are very high. So high it's almost gamebreaking. Almost.

But WX can also eat spoiled foods and benefit from their effects as if they weren't spoiled. Spoiled foods still add the normal amount of sanity/health. Combining this with the "higher max", WX can go on longer adventures, longer treks.

Finally, WX and lightning/rain. Rain damages him, so he's the only character I've build umbrella's for, but lightning is fantastic. You take a sanity penalty, but become overcharged, run a lot faster, give light... Since some lightning strikes are predictable (respawn, hammer the ancient station), you really benefit when needed.

I don't really like Woodie myself, it's a whole different game. I'll leave his story for someone else to provide in an answer.

Wendy has Abigail, a summonable ally who deals area damage. Good at times, but imho a few rock lobsters are better and easier.

Maxwell... well, the high sanity regeneration rate makes him ideal for cave dwelling. His very low max health does mean being careful all the time. Having one meat effigy standing lowers your max health to 45... you'll be wearing full armor (so no backpack) when adventuring! His shadow puppets are fun, especially in the ruins where nightmare fuel is plentiful.