If we already having a center tap rectifier with just two diodes doing the same work then why is there a need to use four diodes or the so called Bridge rectifier.
I'll assume you meant full wave bridge rectifier circuit for "bridge rectifier". To be clear, here is a full wave bridge:
Look at this for a moment and see how it works. It basically performs the absolute value function on a voltage. Actually is looses two diode drops of voltage in the process, but that's not the point right now. If you have a single AC signal, then a full wave bridge is one way to make it all positive.
If you already have the AC voltage coming from a center trapped transformer secondary, then you can use the extra connction to your advantage to simplify the rectifying circuit:
Look at this a bit and see that you always get a positive voltage from V- to V+. So why wouldn't everyone always do it this way? It should be obvious that this second circuit is only possible in limited situations where you have a center tapped transformer output available. If you do, this can be a useful way to do rectification. One advantage is that there is only one diode drop in series with the absolute value of the AC voltage, not two like with the full wave bridge above.
But, think about the cost. Note that only one half of the secondary is conducting at any one time. You are paying for the extra stuff you only use half the time. Diodes are cheap and small compared to transformers, especially at low frequencies like line power. Usually the deciding issue is whether you need a transformer for other reasons anyway, like isolation. In that case the incremental cost of the center tap and winding the secondary with longer but thinner wire is relatively low.
There is another reason for using a center tapped secondary, which is if you want both a positive and negative supply:
Follow thru what happens over a whole AC cycle, and you should be able to see how you get both the positive absolute value and negative absolute value from this circuit.
The biggest disadvantage of the center-tap (CT) transformer is that you only use half of it at each half cycle of the mains. That means your transformer will be twice as heavy and large than one with a full-wave rectifier. The two extra diodes pay for themselves easily this way.
The reverse voltage specification of the diodes is double for the 2 diode verses the 4 diode solution as described in the harshly down voted answer, so this may favour 4 diodes if voltage rating affects their price by more than double such as in higher voltage supplies where around 1500V PIV starts to get expensive.
There are also other reasons for selecting between the arrangements.
The two diode centre tapped solution has value even if one only wanted a positive rail at a specific voltage there might be the desire to have other voltage taps (perhaps selectable) for lower voltages, simultaneously full-wave rectified or perhaps even a simple single diode half-wave rectified rail, these could then share the common ground with the centre tapped rail. It also offers the opportunity to share windings with a ground referenced AC rail (or perhaps even a two phase mains voltage output) that will not be possible as neither of the transformer terminals is at a fixed ground/common voltage when the full bridge method is used for a single rail.
A practical reason for having two totally independent output windings is to allow for series or parallel connection of the transformer allowing use of just two diodes with a centre tap or 4 diodes with the windings in parallel for one voltage or the windings in series with a bridge to select double the output voltage like might be desired in a simple car/motorbike battery charger.
In a microwave oven HV transformer the inner end of the winding is grounded to the core to reduce the insulation requirements (and capacitive loading) between the HV output and the transformer core and to keep the HV-ground at chassis potential thus no diode is placed on the ground side and rectification (or doubling) has to be done on a single grounded winding even though it might be cheaper to do with full-wave rectification a floating HV supply would be unwise especially under fault conditions.
Many practical reasons dictate which arrangement is used and all have up and down sides.
Another thought came to mind. With very low voltages it is prudent to avoid wasting the second diode drop in the diode if it forms an appreciable portion of the output voltage. This might be most relevant in single cell NiCd NiMH charging circuits.
It has to do with the peak inverse voltage of the diodes Please read this article in Wikipedia.
PIV of a diode in a full wave rectifier with centre tapped transformer = 2*Vm PIV of a diode in a full wave rectifier with bridge = Vm
Also, a center tapped transformer would prove to be more expensive than the bridge.