If a tenuto mark means to "hold for the entire value of the note" as it's often defined, isn't this redundant? Aren't we expected to hold notes for their entire value anyway? So what is a tenuto mark truly telling us to do differently?
Tenuto markings often show that a note has extra weight to it. Notes with these markings would not have a sharp/edgy beginning like an accented note, but they are often slightly louder than the notes without tenuto markings. They are also useful in helping to communicate the mood of a piece, as they are more often used in slower, heavier, more somber pieces than in pieces that are quick, light, and cheerful.
There is more to a tenuto marking than that. It can mean slightly different things in different contexts. For example, if there is just one note in a phrase with a tenuto marking, then it would suggest that that note is more emphasised than the others. If you had a row of the same notes, all with tenuto markings, then you would probably play each one with a slight emphasis and therefore making them sound more separated than fully legato. If you had a row of staccato notes and then one that wasn't, you might put a tenuto marking on that one as a precaution. I hope that shows that the tenuto marking is not redundant.
A tenuto marking is somewhat of an accent mark for note length, but to a lesser degree. While a tenuto isn't an outright fermata, it means to put more emphasis on a note. Tenutos used in tempo rubato (free tempo) usually mean to not play the note faster than usual.
A tenuto is somewhat like a swim coach telling you to push hard and swim as fast as you can during a swim meet—you're going to do it anyways, but his reminder serves as extra emphasis.