Terrans
Issues and weaknesses
Thor is too slow and clumsy
Race has difficulties in handling mass zealots in late game
New units and abilities
New "battle mode" for hellion - transforms like a viking
Battle mode hellion has more hit points, moves more slowly
Shorter, wider arc on flame attack, and stronger
Gives terran a beefy front-line fighter in the late game
Warhound
Smaller, more nimble version of Thor
Effective anti-mech ground weapon to help break siege tank lines
Anti-air attack with splash damage
Shredder
Robot that emits channeled, area of effect damage when immobilized
Damage automatically shuts off when an ally moves into range
Gives terran cheap zone control, but can't be used near main army
Is still very work-in-progress
Thor
Moved to late game, more hit points and damage
Can only have one, like mothership from Wings of Liberty
Zerg
Issues and weaknesses
Mid-game difficulties with siege and controlling areas
Missed opportunities with certain units like ultralisk and corruptor
New units and abilities
Ultralisk
Existing version is bulky and has difficulties engaging in battle
New burrow charge ability lets it dive underground and get into the fight easily
Viper
New caster unit with detection -- replaces overseer
Has abilities like blinding cloud to help break entrenched positions
Abduct ability allows it to pull units to itself -- pull siege tanks or colossus out of death balls
Swarm host
Zerg artillery for map control and siege
Burrows into the ground, and continuously generates small creatures to attack
Feels very zerg-y
Protoss
Issues and weaknesses
Needs more raiding options
Needs area of effect, anti-air attacks
New units and abilities
Tempest
New protoss capital ship, replaces carrier
Anti-air splash damage for mutalisks and other aircraft
Has direct air-to-ground attack as well that isn't splash
Replicant
Special unit that can clone itself into any non-massive unit
Allows protoss to also take siege tanks or infestors from enemies
Is very expensive
Still very much a work-in-progress, not-final unit
Oracle
Caster unit used for raiding and harassment
Can disable enemy structures, or prevent miningwith its abilities
Doesn't actually kill or damage anything
Mothership is cut in favor of this unit
There were some other adjustments and additions in multiplayer Heart of the Swarm. Some of the more interesting changes include the reaper gaining the ability to quickly regenerate health when out of combat, though its building attack has been removed. Battlecruisers will also get a new cooldown-based ability for a speed boost.
On the zerg side, banelings will be able to move while underground, through a late-game upgrade, while hydralisks will get a speed upgrade for off-creep movement. Corruptors will also have a new channeled ability called siphon. Siphoned buildings take damage over time, while the zerg player gains minerals as the building is being damaged.
Finally for the protoss, two new abilities have been added to the nexus. One is a mass recall, allowing protoss players to quickly return armies home, while the other is a defensive ability that temporarily gives any structure additional shield and building armor, as well as a weapon similar to a photon cannon.