Looks to me like some independent and assisted living and nursing home centers use amenities to please the families. What are things that matter most?

I agree. Many assisted living and nursing home companies choose to provide amenities to please the family members. It is most important to look at the "guts" of a program specifically whether they've been cited for any major deficiencies, and whether they have a special training program for their staff as well as in-house geriatric psychiatrists and geriatricians. Please look at the recreation therapy prog.
Check out Medicare. Go to medicare url for checklist of important questions to ask http://www.Medicare.Gov/nursing/checklist.Asp rather than be blinded by bling.

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What is difference between assisted living and skilled nursing center and nursing home?

Levels of care. The highest level of care is delivered in the hospitals. Out of the hospital the highest level of care is in a skilled nursing facility that can give complete custoidial care including total patient assistance with daily living, giving medications and intravenous therapy, and all necessary care. The next level of care are assisted living homes that can help people who need less nursing care. Read more...