The first example is grammatically correct to the extent that the subjunctive is [still] a thing in English. Since the subjunctive is a dying form, I will do my best to illustrate my point.
"I hope that" introduces an idea of something happening: a team winning (I hope that the Yankees win tonight), a person recovering from illness (I hope that you get better), it snowing (I hope that it snows on Christmas). We are not speaking about whether these things did/do/will happen, but expressing a vote for their occurrence (an abstract concept). Think of the common phrase, "get well soon!". It's a shortened version of "I hope that you get well soon" and effectively is synonymous with "may you get well soon!". Here you can see that the subjunctive overlaps with the realm of commands.
By contrast, imagine if someone said, "you will get well soon!" Unless the speaker is psychic, this exclamation is rather nonsensical and, as you can see, not at all synonymous with "get well soon". It's something I've heard non-native English speakers say in situations where they clearly meant "[may you] get well soon", and is a grammatical error if used where the latter is called for. Technically, this the same way in which your second example, "I hope you will get better soon", is grammatically incorrect. You can't hope that a person will get better soon. The person will or will not get better; you can merely have your feelings about the notion of one or another outcome. If you are fortunate enough to be God or have a crystal ball, then you might indeed know or predict that someone will get better; "I know that you will get better soon" is grammatically correct.
Does any of that make sense? :)