What is the difference between LTC and NH?

LTC , long term care, is an umbrella term for individual services offered to those needing extended medical care or help with daily living. It can be provided at home or a facility and includes AL, MC, and nursing home care. LTC can also be provided at home.

If your dad is in an ALF now, he may stay there his entire life unless he would require a higher level of care. Your profile says he has dementia, so the next logical step would be a MC facility. Either a self pay facility or a nursing home with MC, depending on his needs and ability to pay.

In NJ and many surrounding states, the term "nursing home" while still used, is somewhat archaic. Many facilities that we used to think of as nursing homes are now a combination of skilled nursing facilities (snf) usually for short term rehabilitation (str) and also have sections that provide long term care (ltc) for people who may or may not need continuing skilled nursing but who definitely need help with at least two of five activities of daily living (adls) which are bathing, dressing, eating, toileting and transferring. Many long term care facilities also have memory care units.
In the US a little bit of funding comes into play also. If there is a medical need for anything you mentioned, you can also private pay for it all. However, if a person is on Medicare, which is a health insurance, be advised that it will never pay for long term care; Medicare pays only for skilled nursing and short term rehab up to 100 days in a benefit period.

My dads skilled nursing center said they are also LTC. I think it might have had to do with the fact that after the rehab was done, he ended up staying for 6 more months so I think at that point, it was LTC.

Different jurisdictions use different terminology.

Here in BC Extended care, Long Term Care, Gentle Care (like memory care, locked ward), assisted living and independent living are common terms.

Long Term Care describes a level of care, but not the location.

Look around and search this site. They are the same thing.

IMO they are the same thing. Both you need to be 24/7 care to get in. LTC just sounds better