Two reasons why it is important to me to use them:
1. Sometimes the characters should known things the players don't.
2. Sometimes the players know things the character's shouldn't.
Since I like to keep in-character and out-of-character knowledge separate, it's kind of important.
The first time they run into something I'll either describe it, show them a picture, or both. Proficient characters essentially get a passive skill check to know basic info about the creature. If anyone (proficient or not) wants to know a specific detail they can make a skill check.
Once a character is exposed to any information (including about creatures) I assume they more or less remember it--especially when it involved survival in a life or death situation. Having proficiency in a skill represents a broad body of knowledge and a good chance to know about anything within that knowledge, but you don't need any proficiency to pick up facts and file them away. So the proficient character can easily fill in the party ahead of time if they are going up against a specific foe.
Lack of specific correlation between skills and creatures types has kind of bugged me. I don't mind letting it be bit more open, but I do mind how the descriptions mention some creatures and not others. Here's my tentative way of splitting it up in 5e:
Arcana: Aberrations, Constructs, Elementals
History: Humanoids, Giants, Dragons, Monstrosities
Nature: Beasts, plants, fey, oozes
Religion: Undead, Celestials, Fiends
The reasoning is that Arcana covers creatures that are strongly magical in nature. Religion covers creatures that are strongly connected to the Upper or Lower planes, or infused with good/evil positive energy/negative energy. Nature covers natural creatures and the fey (especially with the stronger mystical connection to nature that they are given in 5e). History covers creatures that have a society*, as well as those that are most likely to be known about through legends. (I really liked the Knowledge (folklore) skill from some of the earlier playtests, and put a bit of that into History).
Also, if you speak a language, I assume you know something of the culture of the species the language belongs to.
Dragons and monstrosities were the only difficult ones to classify, since they can both be magical to various degrees, but I decided to place the dividing line so as to make the History skill stronger, since Arcana, dealing with all thing magical, is already a more attractive skill than History overall.
Of course, those are your defaults. You can always make a case for using a different skill to know about something.
*I fold the 3e Knowledge (local) and Knowledge (nobility & royalty) into History. Basically anything about peoples and societies.