Vis a vis classical music, may I interject an interesting notion pertinent to this thread. Sir Charles Mackerras, interviewed on a CD accompanying his direction of the Brahms' symphonies (with the Scottish Chamber Orchestra--a marvelous collection), stated it is well known from various students of Brahms that the maestro encouraged subtle variations in tempo. A broadening here at a lyric passage, a touch of strigendo there at a climactic moment, was all fine with him, and not just where he noted it in the score. He was quite opposed to metronome-like adherence to a fixed tempo in the playing of his work.
As a chamber music pianist who plays a good deal of Romantic music, I find this, while desirable, at times difficult. For one thing, if I don't have my part down cold, the string players find it easier to play fast at their first rehearsals. For them, paradoxically, slowing makes bowing more difficult. Especially with Brahms and his dramatic music, the codas tempt them to accelerate, especially given the information above, beyond what my fingers can handle. I'm thinking of the finales of the B Major Trio, the D minor sonata for violin/piano, the E minor sonata for cello/piano, etc.
For me, the key idea is "subtle" variation in tempo. I have heard too many musicians race for the barn at these codas in a not-very-subtle way, and I cannot hear the nuances of the music. And it feels rushed.
Another interesting bit is that Beethoven, as a young pianist, was reported to slow his tempos slightly at crescendoes and climaxes, a practice listeners found original and effective. (He also used a ton of pedal, but those were different pianos, and that's another thread).
Metronomes, I suppose, can be a useful learning tool, if used sparingly (we use one in rehearsal very very rarely, mostly to settle arguments about unintended rushing), but tempos are fluid creatures that can and should be altered, if done with finesse.
As to session playing with click tracks in jazz or rock--that's another animal altogether. It don't mean a thing if it ain't got that swing.