The element linking the measurement and the final control element is the controller. Before the advent of computers, the controllers are usually single-loop PID controllers. These are manufactured to execute PID control functions. These days, the controllers can do a lot more, however, easily 80 to 90% of the controllers are still PID controllers.
It is indeed difficult to say that analogue controllers are definitely better than digital controllers. The point is, they both work. Analogue controllers are based on mechanical parts that cause changes to the process via the final control element. Again like final control elements, these moving parts are subjected to wear and tear over time and that causes the response of the process to be somewhat different with time. Analogue controllers control continuously.
Digital controllers do not have mechanical moving parts. Instead, they use processors to calculate the output based on the measured values. Since they do not have moving parts, they are not susceptible to deterioration with time. Digital controllers are not continuous. They execute at very high frequencies, usually 2-3 times a second.
Analogue controllers should not be confused with pneumatic controllers. Just because a controller is analogue does not mean it is pneumatic. Pneumatic controllers are those that use instrument air to pass measurement and controller signals instead of electronic signals. An analogue controller can use electronic signals. Compared to pneumatic controllers, electronic controllers (can be analogue or digital) have the advantage of not having the same amount of dead time and lag due to compressibility of the instrument.