What is the difference between a solid state relay and a transistor?

One essential feature of a relay, solid state or not, is that the input and output are isolated. In practise this means optical isolation in the SSR (solid state relay) case. In contrast, ye olde phashioned mechanical klunkety-klunk relays are magnetically isolated. One could conceivably make a solid state relay using magnetic isolation in various forms too, but optical isolation makes more sense for the requirements.

So solid state relays are more than just a tranistor, triac, or whatever is used to perform the actual switching. They have an isolated input that then ultimately controls the solid state switch. In practice, this usually means at least a LED and phototransitor in addition to the switching element. That is all packaged together and called a Solid State Relay.

There is another major difference between a transistor and an SSR. SSR is an on/off device, current either flows or it doesn't. A transistor CAN be used as a switch, but in many cases, it operates as an amplifier, ie an extremely small current through the base allows a much larger current through the collector (assuming bipolar transistor). There are actually many different kind of transistors! Bipolar junction, MOSFET, JFET, etc...