Is there an easy answer to the difference between assisted living, nursing homes and care homes?

I have done a good bit of research on only two of the facilities you ask about; I have no familiarity with care homes except that I am sure they are not suitable for a person that needs skilled nursing care. I believe that nowadays what we used to call a "nursing home" is more accurately described as a Skilled Nursing Facility (SNF) and is a place where virtually all of a person's chronic medical needs can be met, though a hospital is still necessary for acute medical needs. SNFs are equipped to deliver many treatments including oxygen and wound care for example and typically have a doctor that visits frequently, perhaps every day. They are highly regulated -- and have different roles and requirements as defined by state law and Medicare (which provides certain coverage in a skilled nursing facility but NOT for services In Assisted Living). People can stay in SNF for short periods, often following a hospitalization for illness or injury so as to recuperate and/or have rehabilitation services. However, I infer from your question that your concern is with long-term stays in a healthcare facility of some sort.
In Assisted Living facilities (AL) residents are much more independent both medically/physically and mentally than is the case for long-term care nursing care. Each ALF has their own standards but will evaluate both the physical and mental state of an applicant to determine if they can provide appropriate services to that individual. For example — and I think this is fairly typical — in my state in order to "qualify" for Assisted Living the resident must either NOT be incontinent or must be able to manage his or her incontinence entirely on his/her own; must be able to feed himself/herself and must have "appropriate behavior" in public places -- no food throwing, hitting, angry outbursts, disrobing, etc. I think these are expected to be quiet, orderly places and, in at least some places/some states, the resident has the right to REFUSE medications or to take a shower or go to a meal. That is, care/assistance is "self-directed" by the resident. Depending on the facility there may need to be a POA on file in order for the AL to discuss the resident with the family. My Mom's facility is, so far, much less "formal" but I think that's because it's very clear Mom doesn't object to them sharing all kinds of info with us; if she objected I'm sure they would not tell us anything unless we produced a POA.
The range of "assistance" in AL varies from place to place but I think typically includes such things as help with bathing and dressing and dispensing of medicines, in addition to including all/most meals, laundry, 24/7 Security (or at least some one on duty 24/7) and light housekeeping as well as regular activities. It may also include transportation to and from doctor's appointments or other personal appointments, usually for an extra fee. Many also have "health and wellness" services such as a beauty salon or fitness center. For Mom dispensing Rx has been the only "assistance" she needed for the first three years but she really did need that and her needs are increasing. Her monthly rent includes Rx dispensing; the AL facility my grandmother was in provided it, but for an additional fee. There were similar differences between the two facilities for what assistance in bathing/dressing was included in the monthly fee. I believe the fees for AL typically are monthly whereas the fees for Skilled Nursing (like hopsitals) are DAILY fees.
MANY people in skilled nursing facilities (which are also MUCH MORE STRUCTURED than Assisted Living or even Memory Care -- more like hospital routines) also have dementia but their medical problems don't permit them to be in Memory Care. Memory Care is a "step up" from AL for sure — more "assistance" in bathing, dressing, "activities of daily living" including being taken to the bathroom and dealing with incontinence, but skilled nursing facilities are the most medically intense and, I believe, the most expensive of any of those about which you have inquired. I'm not going to try to quote you fees because they range hugely, not just from state to state but from facility to facility of each type, as well as the size of the individual's room or apartment. I believe a "care home" is typically less expensive than any of these (but I think not suitable for someone who needs skilled nursing care) but will also have fewer services/fewer staff along with fewer residents. I hope this helps! Good luck on your journey.

The cost for these places are staggering, am I the only one wondering how people do it financially?

Depending on your area and your resources. Assisted Living Facilities offer (almost) total independent care. They do provide meals...laundry...and house cleaning...and a plethora of other services for additional fees. My area this is between $5500 and $6500. A nursing home can be paid for via Medicaid or Medicare (check your area for benefits)..not my first choice as these facilities are horrible at best. Private care homes range (again in my area) $4000 to $6500...depending on location and whether the individual shares a room with 3 other residents or has a private room. Personal care homes are just that...meals...laundry and activities are included in the price.

In my area the independent living facility are about $2300-3000/mo. Meals are extra. Minimal.services. A few places offer activity bus, or scheduled trips to.medical center (good luck coordinating van schedule to available doctor appt).
Assisted living in my state has to include some.meals, and have 24 hr awake staff. A step up from Ind Living......but be aware, Assist Living may have some type of 24 staff just so they can."qualify" to be reimbursed by LTC insurance. They also can evict you, and dont have to have a reason (check all the fine print).
Cost for AL in my area begins about $3500 but most residents pay, after all extra meals, medications help, etc etc, closer to $5500. Per month.
Nursing home is for people who needs much more help with ADL's. Average cost of.semi-private NH is about $90, 000/ year. Plus you still.pay separate for medications, doctor, dentist, eyeglasses, etc. And all your loved one's clothes will.get lost.in the communal.laundry so count on replacing.socks, teeshirts, etc more often than you'd like.

Rejoice 01 I own 2 adult care homes they are better you have more one on one attention and they are cheaper

There's no easy way to shop price. That information is generally only available during a phone conversation. As far as quality? Here's a Medicare site where they rate nursing homes on a 5-star system:

http://www.medicare.gov/nursinghomecompare/search.html

Make a detailed list of your questions for each of them, keep careful notes, and start making calls.

Me? I had mom admitted to a particular nursing home (for rehab) because of a good friend's recommendation. Her aunt has been at this particular nursing home for five years on Medicaid from Day #1. She's 101 years old. My friend said they treat her no differently than they treat their private pay patients. That was enough for me. I always ask friends and family -- but you may want to ask about specific ones after you've gotten information. That'll maybe even generate new names.

The easiest answer is to get the patient in under private payment for at least two years. A lot more doors will open for private pay. If you wait until all assets are spent and you have to go in on Medicaid, you will have very few choices and long waiting lists.