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.
This post is already three years old, but the question "Aren't we expected to hold notes for their entire value anyway?" deserves a more detailed answer than just the simple "No" in one of the comments.
At least in pre-romantic music, notes were expected to be separated by gaps or pauses. IIRC,
G.D. Türk (see edit below for correction) gave the rule that the gaps should take half the note values. The pauses are stolen from the length at the end of the notes. This is called articulation, and varying the lengths of these pauses is one of the most important means in music performing. It is mostly used to distinguish good and bad notes in the musical metre, but can also be used for phrasing: a slur connecting two notes, e.g., often indicates that the second note is to be shortened, thereby creating a sighing effect.
The other important means when performing music is agogics, which means altering the length (including the gap introduced by articulation) of the notes. Quantz (ca. 1750) gave the rule that in chains of the shortest note value in a piece the first of each group (falling on a good time in the metre) should be prolonged and the subsequent note(s) shortened. This is nowadays mostly associated with French baroque music and 20th century Jazz, but Quantz made it clear that it applied to all music.
On instruments without dynamics (organ, harpsichord), articulation and agogics are the only ways to make real music from mere notes. On instruments with dynamics (lute, piano, violin) loudness (dynamics) is a third method, but this does not mean that articulation and agogics should not be used on these instruments.
Edit: Just had a chance to lookup the citation in Donington's "The Interpretation of Early Music" (quite old, but still a very useful collection): It was not Türk, but C.P.E. Bach who wrote 1753: "Notes which are neither detached, slurred nor fully sustained are sounded for half their value." G.D. Türk wrote 1804: "If Notes are to be played in the common way; that is to say neither staccato nor legato, the fingers must be lifted a little sooner than the Time of the length of the Note is expired." The recommended amount of "default articulatuon" thus varied, and might also depend on the instrument: if the sound decays much slower (like on a modern piano compared to a harpsichord or a pianoforte), a stronger articulation might sound less natural.
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.
Articulation of any kind is never assumed. It is always based on the musical context, what is appropriate for that particular moment. When a composer specifically adds an articulation then he is specifying what he has in mind at that moment. A tenuto merely clarifies that “these notes should be full length“, not something else the performer may have had in mind. There is no accent intended, no extra weight implied, no mood, no particular emphasis, etc. If any of those things do happen, it is in no way a result of the tenuto marking, but some other context of the music.
This is, in a way, not an instruction how to play the note but how to make it appear to the listener. Without accent markings, there are some freedoms in articulation, like doing things leggiero. On an instrument with continuous tone control, you wouldn't interrupt tone control (like breath or bellows pressure) for successive leggiero notes but would distinctly separate them (tongueing, alternating fingers on the same key with nary an interruption) and maintain the phrasing.
The notes are not accented on their onset but accent the progress of sound, without merging them into one another like legato would.