"Wherefore, whoso believeth in God might with surety hope for a better world, yea, even a place at the right hand of God, which hope cometh of faith..." (Ether 12:4)

"O ye that are bound down under a foolish and a vain hope, why do ye yoke yourselves with such foolish things?" -- Korihor (Alma 30:13)

Question: what's the difference between a real hope and a false (or 'vain') hope?

"Hope" represents future situations you are looking forward to, and working toward.  The benefits of 'hope' to an individual may involve having a greater sense of optimism and purpose, and possibly a greater ability to endure current struggles with an eye towards things being better in the future.

All of these present benefits of hope are true, though, whether the hope itself is true or false.  Presumably, then, the main difference between real and false hopes is that the real hope will, in fact, be realized in the future, and the false hope will not be.  In other words, the danger of 'false' hopes is that sometime in the future the 'hoper' will be sorely disappointed that what was originally hoped for never materialized--at which point that person may be worse off in terms of sadness and pessimism than if they had never held that hope in the first place.   One can arguably say that the personal feelings of peace and happiness that come from both kinds of hope are exactly the same, only that one will eventually lead to a future period of sadness and disappointment, while the other will not.

It's interesting, then, to hear people who don't believe in God say that belief in God is a 'false hope' because that implies that even though belief in God may provide some tangible comfort to us in this life, we're only setting ourselves up to be sorely disappointed in the future

Problem: if they are right, and there is no afterlife...then when would we have any kind of 'existence' to actually contemplate that disappointment?   If we maintain that hope provides some abstract emotional benefit, in this case is there really a difference between hoping (falsely) for the existence of God and hoping truthfully?   If that 'day of disappointment' never actually occurs, what has one lost in believing in God in the first place?

Let's imagine a parable of sorts.  (Like all parables, this is somewhat of an oversimplification and not 100% applicable to all situations, but still useful...)

There is a passenger jet plane cruising over the ocean.   Things proceed smoothly until the pilot and co-pilot become aware of a severe mechanical failure, such that in ten minutes time, the entire plane is going to explode and everyone inside will perish.  (They decide NOT to inform the passengers of this failure, since there isn't anything anyone can do about it, anyway.)

From the passengers' perspective, they have a 'hope' that they are going to reach their destination, and that their lives are going to continue on for many days and years following. Given the situation, we can classify that hope as 'false' since the truth of the matter is they only have ten more minutes to live before the plane explodes.

The pilots, on the other hand, have no such hope--they have a more accurate grasp of the situation and know what's coming...yet the fact that they do not have the false hope that the passengers do, doesn't make much of a difference to them in the end, does it?  The pilots are still in line to suffer the same fate as everyone else and their advanced knowledge provides no benefit--nor does the lack of knowledge by the other passengers provide much of a detriment.  (Other than the passengers are still calm, peaceful and happy for those ten minutes, while the pilots are probably very, very nervous...

Taking this a little further, suppose the pilots were wrong in their diagnosis--there was no mechanical failure and the plane was not actually going to explode.  Those ten minutes passed peacefully, and the plane continued onward to its destination, exactly as the passengers had hoped for when boarding.

In this situation, it's now the pilots' "hope" (or 'expectation') for the future that proved to be false (although it would be hard to argue in this case that they would feel disappointment at being false).   Despite the difference in hope, in the end the passengers and pilots all wound up in the same destination.  Other than the ten minutes of panic and fear, there really is no difference between the pilots and the passengers due to their difference in hopes.

Let's continue even further: suppose the pilots--AFTER making the flawed diagnosis--make several irreversible life decisions based on the assumption that they have only ten minutes to live.

This could be anything from radioing their supervisor to tell them exactly what they think of him, stripping off their clothes and running naked around the plane, you know, just for the heck of it, or perhaps throwing open the emergency exit and jumping into the ocean, preferring perhaps to die in the ocean via skydive versus passively waiting for an explosion. (In this case, as always happens in movies, let's assume the passengers just happen to include a retired pilot or military person who has the expertise to guide the plane to a safe landing, even without the pilots)

In this case, there is a great difference between the pilot and the passengers since one made significant decisions based on his 'knowledge' that life was not going to continue beyond point X, while the others did not.

In all three cases, regardless of whose hope was real or false, the passengers' eventual circumstance ended up being the same or better than the pilots' every time.  Compare this to believing in an afterlife which turns out to be false--what difference does that make in the end?  (As compared to falsely *not* believing in an afterlife, which can make a difference...)

Now, it would be cynical to say the lesson is: believe in God whether you believe in God or not...just to be safe.  It does present the question, though, of what exactly the harm in 'religion' in general is?  Atheists have a higher burden of proof that belief in religion--even if false--actually harms the believer, versus them being either in the same or better position than the non-believer in the end, anyway...