What is difference between wheel balancing and alignment?

The alignment is of the actual rotor and suspension system  so that your vehicle tracks properly and that your wheels will be true both in direction and camber.

A Balance is to be sure that your wheels are properly weighted to prevent shudder.

I balance every other tire rotation, and align about every 60,000 unless I hit a pot hole. Also check your tires for even wear, if they are wearing out unevenly it likely because of alignment. If your car is shuddering at speed its likely a balance issue. 

Balance is usually 10 to 20 bucks per wheel and, alignment, if there are no worn parts will usually be between 150 to 200 dollars.

When they balance the wheels, they are making sure that they will not bounce or wobble at high speeds.

An alignment means they do their best, within specifications, to align all four wheels so they are all pointing precisely forward. Bumpy roads, potholes, and regular wear and tear on the ball joints, tie rods and other parts can adversely affect the correct alignment of the wheels and can cause the car or truck to pull to one side of the road and can wear out the tires prematurely. 

I've seen cars where the wheels wear so far out of alignment that there was steel belt showing on the tires three months after they bought the new tires. 

(And they expected to get four new tires for free) 

wheel Ballance is done off the Car on a Machine

Alignment is done on the Steering Part of the car

where Caster Camber and Towin is checked and corrected where Necessary

Wheel rims are round "yes they are"

Tire construction is a flat band of rubber, and a layer of fiber, and a layer of steel and another layer of Rubber tread and then they chop it to length and put inside a round oven press and heat it up and then press it into shape.  Find it on YouTube   Making car tires and watch video really interesting and you will never ask again.   But as the ends must overlap then you have a point where the rubber is thicker(or heavier)

Lead weights are added to rims to counter that heavy spot with lead so the tire rolls without vibration going into the steering wheel.  All 4 wheels are balanced.(because many people rotate their tires.  With radials it is front to back and back to front same side.

Wheel alignment...look at your feet. are you pigeon toed or have the cowboy been on his horse too long stance. Pigeon toed means tires are / \ (or toed in)& cowboy is \ /.  You want the wheels to be slightly pigeon toed because when you travel at a high rate of speed the wheels straighten out.   This is all done on a wheel alignment machine and by lazer alignment.  All wheels are aligned.  The guys know what they are doing.  Car will drive straighter and corner better with the steering wheel automatically returning back to straight after you make say a sharp right or left and then let the steering wheel rotate on its own between your hands as it tries to straighten out on its own.

. This gives the least amount of tire wear.

When it is toed out it won't try to return to straight driving.  When you angle park do not slam into the curb as the parts are only STEEL.  You can break steel and the wheel alignment shop will also check for worn out steering bushings or struts or leaking shocks.,  They know the underside of the car better than any engine mechanic.  Generally they want you in and out, there are other cars that need alignments.  They don't do exhaust or tires.  They specifically do one job.

Wheels are not completely round,  yes really! ( well the weight distribution is not perfect).

Balancing puts weights on the rims to compensate for heavy spots and stops wheel wobble.

Alignment sets the wheels at the correct angle , rarely parallel to each other , normally a degree or two pointing in or out depending on how the suspension is designed. Wrong alignment will give inaccurate steering and wear the tyres out rapidly.

One balances the wheels and one aligns them

Tire balancing is where the wheels are removed from the vehicle and put on a spin balancer,  the machine detects where and how many ounces to apply the wheel weights so the wheel will spin smoothly.  Alignment is where the vehicle is put on a machine and alignment instruments are affixed to each wheel,  they check for camber, caster, tow in, and the the cases of vehicles that require a 4 wheel alignment, it is checks for "dog tracking"  that is where the front and real axles are aligned with each other, this is to ensure the tires to not wear out prematurely.

Wheel balancing is where the tires are taken off the vehicle and spun on a machine to determine if they are out of balance which will cause vibration while driving, 

When you het an alignment the vehicle is put on an alignment rack that checks the camber, caster, and toe adjustments of the suspension. When you buy new tires the tires always need balancing. The old tires should be inspected for signs of alignment wear so that you know if you need to get a 4 wheel alignment so your new tires don't wear out prematurely.

Well, William . . . . my guess would be that one procedure involves balancing the wheels and the other involves aligning them.

I notice that this is your 20th question about tires in 10 months . . . . . got a little problem there, don't you?