For a while I thought they were one and the same, now I'm confused. This Wiki Article on the subject asserts that the blue note isn't even a real note, i.e. a note that's not part of the standard scales. Please elucidate this for me.
There's no standard precise definition of what a blue note is. In some cases, a blue note could be seen as a deviation from the pitch of the note of a 'standard' (e.g. major) scale, but in other cases, blue notes are used in music whose tonality is far enough removed from the major or minor scale that those scales are arguably not sensible reference points to measure deviations from.
The quote in the Wikipedia article seems quite good:
Like the blues in general, the blue notes can mean many things. One quality that they all have in common, however, is that they are lower than one would expect, classically speaking. But this flatness may take several forms. On the one hand, it may be a microtonal affair of a quarter-tone or so. Here one may speak of neutral intervals, neither major nor minor. On the other hand, the lowering may be by a full semitone--as it must be, of course, on keyboard instruments. It may involve a glide, either upward or downward. Again, this may be a microtonal, almost imperceptible affair, or it may be a slur between notes a semitone apart, so that there is actually not one blue note but two. A blue note may even be marked by a microtonal shake of a kind common in Oriental music. The degrees of the mode treated in this way are, in order of frequency, the third, seventh, fifth, and sixth.
— Peter van der Merwe, Origins of the Popular Style, 1989
We can see from that explanation that blue notes are not simply the use of flattened notes that themselves still belong to the 12-tone scale, as blue notes can be quarter-tone deviations, or characterised as a glide though a frequency range.
Although the idea of blue notes is not restricted to the blues scale, this question - Exact pitch of blues scale notes - has some relevant answers. In particular, you can see from my answer there that I think that there's a note in the blues scale that is - in a guitar context, at least - played as a raised fourth more than a flattened fifth, though it may depend on the style in question (and of course such a distinction would be meaningless on the piano.)
When talking about 'blue intervals', we normally mean intervals from the root - so the 'blue third' is some kind of (altered) third above the root.
A tritone is simply an interval of three tones. So we can say that a note an interval of a tritone above the root could have the same frequency as a 'blue note' played between the fourth and fifth. That's not to say that a tritone above the root is a blue fifth though - blue notes are more complex than that.
They are two different concepts completely, although the blue notes do typically take advantage of the tritone via harmony and melody.
A tritone is a specific interval of an augmented fourth (A4) or a diminished fifth (d5). It's a very special interval as it's the furthest distance in semitones you can be away from a note and it's dissonance and how it resolves is very central to a lot of compositions. A simple example from C to F♯/G♭ is a tritone. This interval is an essential part of the dominant 7th chord as the interval between the major third and the minor seventh in the chord is a tritone.
This leads us to the blue notes. Alone it's just notes to color the melody and harmony of a song, typically in a slightly dissonant fashion. The blue notes are ♭3, ♯4, ♭5, and ♭7 that are deviations from the typical major scale. From these notes tritones can be and are typically formed between other notes in the melody and harmony, but the notes themselves are not tritones. The ♯4 and ♭5 create a tritone with the tonic note. The ♭7 makes a tritone between itself and the 3 like in the dominant 7th chord. The ♭3 makes a tritone between itself and the 6 which is also root note of the relative minor of whatever major key you are in.
However, the blue note is not just used for there tritone properties, but also other properties as well including chromatic. It's pretty much standard to hear either 3, ♭3, 2 or 5, ♭5, 4 in a blues lick.
So in short the blue notes are notes added for flavor or color that can produce a tritone, but don't need to while a tritone is a well defined interval.