They're the same thing, on a low enough level.
In semiconductors, electrons are promoted from a bound (valence) state, to a conduction state. The energy change is the band gap, a few eV.
Unique side effect: the vacancies (holes) also move, with somewhat poorer mobility (in silicon, about 2.5x slower -- which is why PMOS have to be bigger than NMOS to make a complementary (CMOS) circuit; most semiconductors have a much greater ratio, making CMOS impractical in anything else).
Semiconductors have a bulk charge balance, so the charge carrier density can be extremely high. Downside, the bulk also obstructs flow, causing thermalized diffusion motion. Since the junctions can be made very small (indeed, must be), this relatively low velocity only has a limiting effect at high frequencies.
Also, the energy distribution of those electrons, given by Fermi-Dirac statistics, has an exponential cutoff. Which is essentially why semiconductors have an exponential response. (Diodes and BJTs, and MOSFETs in the subthreshold region; FETs above subthreshold are another matter, which I forget why at the moment.)
In vacuum tubes, electrons are promoted from a bound (valence) state, to a free (vacuum) state. The energy change is the work function of the cathode, a few to tens of eV.
Electron emission can be stimulated by simply heating the cathode, or by irradiating it with light of adequate energy (photoemission) or other particles (secondary emission).
Only electrons can be emitted in this way. Probably, with some work, a "P type" tube could be made, using a hydrogen fill gas and a hollow cathode to emit ions (protons). Unfortunately it would have utterly terrible performance (proton mass ~2000 times the electron!).
Vacuum tubes do not have a bulk charge balance, so the electron density is low, and this is forced by the space charge of those electrons in the beam. (That is, electrons repel, so a cloud or beam of them tends to push back on itself.) With no obstructions, the beam can reach high velocity -- ballistic transport. Unfortunately, though high voltages can be used, practical reasons limit how closely electrodes can be spaced, so the effective transit time is worse than most semiconductors.
Electron energies are given by the Maxwell-Boltzmann distribution, which has an exponential tail. Vacuum tubes are also exponential in the cutoff region; this is a relatively small part of their operating range however, and dynamic range isn't that great to begin with (due to the high temperatures, and construction and some chemistry details, leakage currents are relatively high). In the normal operating range, a 3/2 power law (give or take exact electrode geometry) occurs. (While this is more linear than an exponential transfer function, it's also much lower gain -- on top of the already low gain due to the low charge density in the vacuum.)
The peculiar characteristic of the triode (and to a lesser extent, multigrid tubes as well) is a low plate resistance; this is not fundamental to the mechanics, so much as a consequence of the electrode arrangement. Some plate voltage leaks through the grid. The grid acts as an electrostatic shield, but a poor one, so there is a ratio of influence between grid and plate -- the amplification factor, µ. This applies for all electrodes, so that in a pentode, the suppressor grid partially shields the plate, then the screen and grid in turn as well. So the µ from plate to grid can be quite high, but it's just the product of each grid's µ to the next. (This is different from FETs, which have a mechanism that effectively shields the gate and drain. A different, much weaker effect dominates: channel length modulation.)
So to a certain extent, vacuum tubes and semiconductors already are the same things.
To a more meaningful extent -- if you want a semiconductor that acts like a triode specifically, there's this:
https://en.wikipedia.org/wiki/Static_induction_transistorLike the triode, the electric field inside the junction happens to influence current flow, thus giving a low output impedance. (They aren't very available, if they're in production at all?)
Others have covered the more traditional interpretation (substituting semiconductors into a vacuum tube circuit) so I thought I would talk about something more fundamental.
Tim
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