Are solid fuel rocket engines designed differently and how?
Yes, solid fuel rockets are very different from liquid fueled rockets. A solid fuel rocket is basically some solidified rocket fuel and oxidizer wrapped in a non-combustible cylinder (hence the name). Once it's lit, there's no way to throttle it or shut it down, it just burns until all the fuel is consumed. On the other hand, in a liquid fuel rocket (cryogenic or not), liquid fuel and oxidizer are pumped into a combustion chamber to be burned. Liquid fuel rockets can be throttled or shut down by controlling the fuel pumps, and can be re-lit once shut down (assuming its igniter is reusable).

(It's important to note that the examples of "solid" rocket fuels you gave aren't actually solids. They're liquids, just like cryogenic fuels are. A solid fuel would be something like the ammonium perchlorate and aluminum powder used in the Space Shuttle Solid Rocket Boosters.)