Impossible Difficulty (EU2012)

Making the transition from Classic to Impossible can be a steep climb. Here is a comparision of the main differences between both difficulty modes.

Tactical

XCOM soldiers start with base 3 HPs (and +1 from Body Armor), a reduction that will have effects during all missions but making them very susceptible to KIAs from Sectoid Plasma pistols at game start.
Aliens will deploy on bigger numbers (usually 1 additional pod) during the 1st mission and specially on Terror Sites/UFOs. It can be a challenge to know how many pods are still inactivated.
The AI will use grenades more, specially when facing Muton Elites or Heavy Floaters. After the aliens start fielding grenades the full cover positions in most maps will be very risky to use, since it will be blown away.

Sectoid HPs are increased to 4 HPs, preventing them from being killed by single Frag Grenades.
Floaters and Thin Men HPs are increased from 4 to 6, making Assault Rifles even more unreliable to kill them, however, their Aim remains the same as in Classic. Rocket Launchers still retain their 'kill with 1 shot' ability against both aliens.
Outsiders have 90 Aim, 10 Defense and 17 Mobility (the same as Bezerkers). Be very careful with activating Outsiders on Impossible by mistake - a single one on the loose will wreck havoc amongst your troops with their high Aim and Movement.
Chryssalids remain the same as Classic but Zombies are the bad news. They have 14 HPs (10 in Classic) and their Damage bonus is increased by 4 (+2 on Classic).

Muton and Cyberdisk stats remain the same as in Classic.
Drones HP is increased to 7 HPs on Impossible, 2 more than in Classic, turning them into a bigger nuisance. It also prevents Drones from being killed in 1 shot by suppressing them with a Heavy Plasma with HEAT Ammo and Mayhem.
Sectoid Commanders HP is increased from 10 to 14 HPs on Impossible. They also have a 35 Will bonus (25 on Classic), allowing them even better odds on Mind Control attempts.
Bezerkers health is upgraded 25% from 20 HPs to 25 and they deal 2 extra points of Damage on Impossible (for a total of 11 base damage).

Muton Elites have +10 Aim (for a total of 100 Aim on Impossible), giving the best accuracy on the game. In addition, their health is increased on Impossible from 14 to 18 HPs, while their Defense remains at +30. This combination of high stats, their Heavy Plasmas and their numbers (just facing 3 can be a nightmare) makes them the most dangerous aliens on all situations.
Heavy Floaters health is added 2 HPs to a total of 16. Their mobility and bigger usage of Alien Grenades (3 tossing grenades at one soldier with deal 15 points of damage) makes them also very dangerous.
Both Elite Mutons and Heavy Floaters now can't be killed with a single plasma shot that isn't ability enhanced (i.e. Headshot, Shredder, Bring Them On) since it will deal a maximum of 15 damage with a critical hit.
Ethereals have +5 HPs and a 35 Will bonus (25 on Classic), allowing them better odds on Mind Control attempts.
Sectopods stats remain the same as in Classic. However, you'll face more of them during missions instead of just 1. Sectopods also seem to use more their Cannon to blow terrain.