The voltages and currents in vacuum tube circuits are naturally much higher than what you'd find in a transistor-based circuit. So a vacuum tube circuit is much hardier when it comes to the high voltage spikes of EM-pulses from nuclear attacks on one hand, but there are other issues with transistors on radiation that vacuum tubes don't care too much about.
A transistor is based on a carefully doped semiconductor. Nuclear radiation can take several forms, alpha, beta, gamma, and neutron radiation. Alpha radiation is fast helium nuclei, so two protons and two neutrons. It's blocked by basically a piece of paper, so not much danger here, although absorption in a material can lead to bremsstrahlung/x-rays. Beta radiation is fast electrons/positrons (beta- and beta+ radiation). Again, doesn't penetrate very deep, but creates bremsstrahlung and, in the case of beta+ radiation, the annihilation of the positrons also results in gamma radiation. Gamma radiation is highly energetic EM radiation, basically light with a very short wavelength, shorter than x-rays. They penetrate very deep and can't be fully blocked. Gamma radiation happens at every nuclear decay that results in alpha and beta radiation because after an alpha or beta decay the resulting nucleus is in an excited state, and when this excited state decays the nucleus sends out a gamma quantum. Neutron radiation then is mostly found in nuclear reactors, and it's, well, free neutrons. They're especially annoying because they can turn stable elements into radioactive elements and cause material failures.
And neutron radiation is what's really the killer for semiconductor electronics, because the neutrons will mess with the doping in the transistor and destroy it. Also, gamma radiation can ionize a transistor locally and scramble its operation.
Vacuum tubes don't care much about gamma radiation, but prolonged neutron radiation will also destroy them by making the glass brittle and corroding the wiring, I guess.
EMP weapons shouldn't have too much effect on Fallout-universe computers and robots, either, actually