The simple answer is that a guitar tube amp uses one or more vacuum tubes to amplify the signal, while a solid-state guitar amp uses solid-state electronics (diodes, transistors, etc.) to amplify the signal. On paper and in theory these two approaches should yield identical result, but in actuality the difference is usually quite noticeable.
The all-tube Fender Hot Rod Series Blues Junior 15W Combo Amp generates warm, musical distortion at relatively low volumes.
But the simple answer fails to address underlying complexities. Many amps are not simply tube or solid-state, but mixes of both kinds, called "hybrids." This usually means that they have a tube preamp stage, employing vacuum tubes in the tone shaping circuitry, but use solid-state circuitry for the power section. The hybrids are closer to full tube amps in response and tonal warmth, but purists will still find a difference between the two. Tube amps are generally more expensive in initial cost and to operate (because you need to replace the tubes occasionally), and solid-state amps are generally less delicate and more reliable. Many players, however, feel that tube amps yield a warmer, more musical tone and more musical-sounding distortion.
Yet another wrinkle is tube emulation circuitry. Many amps and preamps have sophisticated circuits designed to act like tubes, and as in all things, some are better than others. A relatively new development has been the introduction of modeling amps, which not only emulate the tone and response of tubes, but of specific tube amps. There are a variety of offerings available, including popular choices such as Line 6's Spider line and Roland's Cube series. In general, modeling amps are exciting, versatile amplifiers, but again, some are better than others at reproducing the specific models, and in maintaining the sounds through a range of volume levels.
The Line 6 Spider V guitar amp offers more than 200 amps models, cabs and effects to choose from in designing your guitar sound.
Another point to make about tube amps is that bigger is not always better. You get the most distinctive tube sound when the amp is cranked up enough so the tubes are saturated or nearly saturated, creating the overdriven sound revered by tube-amp fans. For this reason, it is often better to choose a lower wattage amp over a higher wattage amp, depending on how and where you play. By the time you crank up your 60- watt amp enough to saturate the tubes to get just the right level of distortion, you could be blowing your audience out the back door. It might have been better to choose a 20W amp that lets you get your saturated tone without the ear-killing decibels. Many professional guitarists prefer this approach both for recording and performance situations. They use close-miking to capture the overdriven sound of smaller tube amps, sending that signal to the recording console or the PA mixer.