Continuous means it is implemented with analogue electronics. Discrete means digital. The results may be the same, the difference is in the implementation.

For the discrete-PID controller you should take the sampling time into account. There are some approximations to convert continuous time PID controller to the equivalent discrete time PID controller. Following approach might work:

Consider a continuous PID controller expressed by the transfer function of

D(s) = Kc [1 + 1/(Ti s) + Td s]

In the discrete-time domain,

Integrator -> T * sum(e(nT))

Derivative -> [e(nT) - e((n-1)T)] / T

Note that it actually changes depending on which type of approximation is used for the integral term (such as forward euler or trapezoidal etc..) . Then by using z transformation,

D(z) = Kc [1 + (T/Ti) * 1/(1-z-1) + (Td/T) * (1-z-1)]

D(z) = KP + KI / (1-z-1) + KD (1-z-1)

As a result, discrete-time PID parameters in terms of the continuous-time PID parameters;

KP = Kc

KD = Kc * (Td/T)

KI = Kc * (T/Ti)

You can find more information in digital control systems books.

There is more to the continuous and discrete-time control than just implementation difference. First of all, a continuous-time controller is designed in the s-domain, whereas a discrete-time controller is designed in the z-domain. Therefore you cannot use the same numeric values to obtain the same response (rise time, settling time, etc)

To convert from s-domain to z-domain, a common approximation is zero-order-hold approximation. For example in Matlab, you can use c2d function, i.e. sys_d = c2d(sys,Ts,'zoh'), where Ts is the sampling time.

However, keep in mind that you need to convert the plant itself to z-domain, not just the controller. Even then, the response will not be exactly the same, but for high sampling rates, the approximation is fairly close. 

For more information see: http://ctms.engin.umich.edu/CTMS/index.php?example=Introduction§ion=ControlDigital

@Yasar small doubt, i will convert the output of controller in to continuous and i will fed them to the  plant. Then, there is no need to convert the plant to z domain? 

A digital PID controller can offer a lot more functionalities than analog (pneumatic or electronic) PID controllers e.g. auto-tuning function, software structure selection (cascade, series, parallel, interactive, etc.) and remote networked configuration. A PID implementation has to cope with stability, robustness and performance. Beyond design in Laplace Domain or discrete Z-Transform domain, an algorithm has to deal with adequate sampling in a digital form, positional or velocity implementation, process wind-up, bumpless transfer from manual to automatic, bumpless parameter tuning, avoidance of derivative and proportional kick, quantization effects in integral action, etc. PID implementation, analog or digital, carries a lot of tricks and although may look as a simple easy task, it is not! Some of the best authors’ references describing how to implement an industrial PID algorithm are Gregory Shinskey (former Foxboro Co.), Karl Astrom & Hagglund (Lund University). Search the Internet for PID books from these authors and you shall be in good hands. Be aware of didactic textbook/handbook PID algorithms that are only good for teaching. Most Companies don’t publish their actual implemented algorithms because this is yet a good source of income.

This question actually represents the dilemma of any control designer. Unless the plant itself is digital, one usually deals with continuous-time systems and implements discrete-time controllers.

So, you may first design the best continuous-time controller for the continuous-time system and then digitize the controller, thus ending with the wrong controller for the right plant.

Or, you may first digitize the plant and directly design the appropriate controller, only now you end with the right controller for the wrong plant.

The ‘best’ would be using hybrid systems theory, which take into account both time domains, yet the results so far are pretty limited.

The best practice (while using the best knowledge from the references) maybe is working in the continuous-time domain, while also using the highest sampling rate you can afford for the controller, so the digital representation of the plant is a good representation of the continuous time-plant.

I happened to work in very fine motion control, such as bonding machines for integrated circuits, where a 3 microns oscillation may mean 2 microns too many. When people would ask me how much computing power I wanted, I would answer:100 times more than you think now. In the beginning it was taken as joke, yet to keep on with the continuously increasing required speed and accuracy, what started with sampling rate of 1 kHz, moved to 4, then 8, 16, 32 and ended with 64 kHz.

to explain it briefly, an determining parameter is the sampling time. depending on the sampling time, the discrete PID coefficients can or cannot be the same as the continues PID coefficients (which mostly they are not equal)

You can find a lot of interesting answers about numeric control in Ioan Dore Landau's books and papers (e.g. Adaptive control or Digital control systems http://www.slideshare.net/BeneditoCurvina/ioan-dor-landaugianlucazitodigitalcontrolsystemsdesignidentificationandimplementationcommunicationsandcontrolengineering2006 ).

well the performance depends upon whether the PID is designed via emulation or in direct digital configuration. In any case the most important factors determining the performance of the PID or any other discrete time controller are 1 ) the sampling time 2) the numerical integration technique (forward rule, backward rule, trapizoidal rule, bilinear approximation or directly by z = exp(sT)) 

Continuous PID can be implemented using electronic elements such as R,L,C, transistor, OpAmp and so on. To rearrange the purpose of the PID we need reassemble the circuits. In other hand Digital PID is implemented using micro controllers, computers so on basing on programming tools. It means that processing numbers only. Therefore it has great adventures then continuous case. We can process the numbers as we want.

Continuous-time controller is designed in the s-domain, whereas a discrete-time controller is designed in the z-domain. Therefore you cannot use the same numeric values to obtain the same response.

Continuous is implemented with analogue electronics. Discrete means digital. The results may be the same, the difference is in the implementation.

For more details, we have:

https://ieeexplore.ieee.org/document/8340444?denied=

Best regards