You say something exists.  You must be clear on what that something is.  Sometimes the declarations about God are so foggy that you can neither deny them or accept them.  You need to know exactly what something is before you can say it exists.

If you believe in God because he cannot be proven or well verified that is not God you believe in.  It is something in your own head even if there is a God.  It is an idol.  So in a sense failing to prove or verify God means you can prove the non-existence of God for all you have left then is an idol God.

Proving a negative is trying to prove that something does not exist or is never true.  Many say you cannot prove a negative.

Trying to prove a negative then is impossible either in itself or just for us for we don’t have enough information.

There are things proven and unproven that we will never know of. Proof is about reality not what we think or know.

Let us think about how we deal with proof? How do you show that a man called Ajax in Siberia with five fingers on the right hand in 5000 BC did not exist? He might have done though. What about the magic god on the moon who lived for 1 second in a million BC? Or the invisible and undetectable man on Mars? We can say they might have existed. The might for Ajax, and similar cases, is stronger. Why? How? Proving a negative is impossible for proof is only about what exists. However if proof that Ajax did not exist exists then

You can prove something exists. That is positive proof. A single proof is enough. When you can prove something does not exist - that is negative proof. Its problem is that you cannot really prove there is no such thing as unicorn living on the sun. So it is not informative. Negative proof can be informative sometimes. Here is an example. You cannot prove that a unicorn with magic ability to survive the sun is rubbish. You can prove that a unicorn as normal as kitten (and that no supernatural power is protecting) cannot be making its home on the sun.

Even you cannot prove there is a magic unicorn on the sun you can still give some reasons, even thin ones, for holding that it does not exist. In the absence of proof for “x is not real” you can still find evidence and good reasons for saying that it is not there.

Negative proof is as much proof as a positive proof. If John never existed that is much a truth as that John does exist. It gets tricky for proof can exist that x does not exist.

Proof is proof and it is theoretically possible for there to be proof that something does not exist. You can have reasons for holding there is such proof though you do not have the proof for sure. You may hold that God can be disproven though you or nobody can do it. It does not mean there is no disproof.

A negative atheist is a person who says they do not believe there is no God but simply do not believe there is one. To show the atheism side more, the negative atheist should add, “I believe there is proof that there is no God though I do not have it.” That is not the same as saying there is definitely no God. It is faith that you have no reason to believe in God and that God can be disproved.

What use is believing there is proof that there is no God even if you will never get it? It subscribes to atheism in all the essentials. The important thing is opposing God as an explanation and as something that has the right to tell you want to do or to support a religion.

To say there is no God is to assert there probably is no God.  All assertions, saying something does not exist, is an assertion too does not claim to be 100% proven but claims to be probably right.  To say you cannot prove a negative is to make a positive assertion (making a claim) that something is probable - eg that there is probably no God. Every position, "X exists", or "X does not exist", is an affirmation of belief.  You are saying you believe something.

Does that mean then that with denial of God's existence or the mere lacking belief in God that there is no meaningful difference?

You can prove the negative for proving does not mean total certainty but is about what is probably right.

The one saying there is a God seems to be the one carrying the burden of proof not you if you say there is no reason to believe in God.  But in fact if you say there is no reason then you get a burden of proof by your own.  You cannot stand by and expect the other to do all the work.  If you are just going to listen then the burden of proof then is on the other person for you are not claiming anything to them.  Not all burdens of proof weigh the same.  In fact if you and the believer have a burden of proof the believer has a heavier one for your claim is more modest then theirs.  John saying there is a tooth fairy needs to do more work than the person saying they have no reason to believe.  The person is not saying there is no reason but making the modest personal claim that they see no reason. God is the biggest claim of all so saying God is real involves having the biggest burden of proof of all.  Remember at all times that burden of proof really means burden to show evidence and good reasons for holding the position.