What simple neurological at home tests differentiate dementia from depression in a senior refusing to seek medical advice?

Could be both, and agree there could be a refusal to interact because he may be aware of the losses--doesn't want to talk because it may make it clear how he's affected, or it could be apathy. He needs a physicial to rule out physical causes too, like a UTI, vitamin or thyroid deficiencies, etc.
Even if you suspect it was one or the other you'll want him to see a practicioner. If you can't get a physician to make a house call you may have to take him, without telling him where you're going first. You would want to communicate with the doctor beforehand what behaviors you see (either by online health portal or a note slipped to the staff at check in.). Medicare does require a yearly wellness check, so you could use that as the excuse for a visit.
Heres, a link to the online MMSE, but be aware that people can 'learn' the test and pass it, so it might be better to have a physician administer it. A true neuro/neuropsych assessment will take about 2-3 hours.
https://www.dementiacarecentral.com/mini-mental-state-exam/

12_pt_Understanding_the_Dementia_Experience.pdf
https://www.youtube.com/watch?v=u5QMeQpkPhA
https://www.youtube.com/watch?v=0BlZF_4EKp4
https://tamcummings.com/stages-of-dementia
https://www.communityresourcefinder.org

I would definitely make an appointment for a neurologist.
In June 2017 my husband was driving (we were in New Mexico heading to Colorado) and he asked me where we were. I became very alarmed. We returned home after a week and I continued to pay attention to his behavior. When we did our 6 month wellness check up I asked our doctor for a referral to a neurologist. I explained why. He said it's just age (my husband was 80 then). He said he needs to learn an instrument or a new language! I was outraged. We did get a referral. Neurologist did tests, had blood work done, etc. He was a negative 7 with vitamin B-1. This is thiamin which helps with your gait and memory. They prescribed 500 mg per day. Eventually his levels were in the 300 range. Now he takes 100 mg of B-1 daily. He also takes B-12, And a prescription for memantine and donepazil which he takes daily (as long as I see that he gets them.) He was diagnosed with dementia due to alcohol, which he used to drink daily.

It is quite possible he knows he is aging and developing problems and simply refuses to accept them or can't - who could. I would find a way to make arrangements with the medical people to do an evaluation - there are ways, you just have to find them. Then go from there. And set some boundaries and if he does not cooperate, tell him you will place him. Good luck.

I agree about calling his insurance to ask about in home visit and/or the "new" Medicare requirement to have an annual wellness exam. There could very well be both depression and dementia, coupled with metabolic issues. Unless he falls or is unconscious 911 is not the best course. Don't give up on him. Does he have any friends or other family members who will help you? Good luck.

You can do an in home test for "Potential" cognitive changes. It's called the MMSE or Mini Mental State Exam. You can find the test online and the scoring criteria.

Seems your father has depression. Rather than "ask", tell him you are concerned about his health and make an appointment for him with a geriatric psychiatrist.

Call his Heakth Insurance and see if you can have someone come to the house to give him a check up.

Sounds like Depression to me.

Also check with the Pharmacists and his Dr regarding all RX meds he's taking, especially the last Rx that was added before he got like this.

TRyan getting him to go out to lunch, go for a drive to the Beach, A Pic Nic at the Park.
Ask him to go with you to a favorite place of his.

Inite a Friend to come for dinner and let them have some private talk time

He's awfully young to have given up!

I agree that he should have blood work done to see if he is low/high in something that could be causing this. See if his doctor will order blood work and then get him to the lab. I'm sure the doc won't write a prescription without seeing him but the results could be enough to get him to an appointment.

He might need to get treated for depression and see if it helps. There is no definitive test, that I know of, to determine if a person is actually depressed.

Good luck.

There are no tests you can do at home that will determine this. This would need to be done by a Neurologist.

Dad would feel so much better if he was checked out. It could be physical not mental. Low potassium can do this. Dehydration. Diabetes. Thyroid. He needs labs done.

IMO calling 911 will not work. If Dad can say NO he doesn't want to go the EMTs can't force him.

You know, he could have dementia AND be depressed.

And if you administer an at home "dementia screening", what will that gain you?

Have you tried telling him that he will lose his Medicare insurance if he doesn't go to the doctor for a once a year check up (this is a common therapeutic fib that many of us have used to get resistant loved ones to the doctor).

If anything seems amiss--a fall, confusion, delusions or hallucinations--call 911 and have him checked out in the ER. Make sure you have a list of your concerns handy and on you at all times.