What is the difference between CAN and ARINC?
the communication between the cockpit and the controllers is done by using Arinc protocols. And the data transmitted by probes to the controllers is done by CAN.


That depends very much on the aircraft type. 


  And when can we use each one of them (Arinc and CAN)


When you design your own aircraft you get to choose.



CAN stands for Controller Area Network. CAN is a bus standard designed to allow microcontrollers and devices to communicate with each other in applications without a host computer. It has its origins in the automotive industry. 

ARINC stands for Aeronautical Radio, Incorporated. ARINC is an organisation that amongst other activities produces technical standards for avionics and provides services to aircraft operators. 

Amongst the ARINC standards there are a number addressing data communications between avionics components. Most widely used is the ARINC 429 standard which describes an avionics data bus. ARINC 429 allows a single transmitter to communicate data to up to 20 receivers over a single bus. 

Comparing CAN bus to ARINC 429:


Cable


CAN: STP or UPT, possible to implement on fiberoptic cable
ARINC 429: STP

Speed


CAN: 1 Mbit/s (high speed) / 125 Kbit/s (low speed)
ARINC 429 100Kbit/s (high speed) / 10-14 Kbit/s (low speed)

Directionality


CAN: bi-directional, connected units can send and receive on the same cable. 
ARINC 429: uni-directional, a unit either transmits or receives from a cable.

Number of units on a single bus


CAN: unlimited, although in practice no more than about 32 units will be sharing a bus to avoid data collisions.
ARINC 429: one transmitter can serve at most 20 receiving units on a single cable.



Since ARINC 429 has only a single transmitter, timely data arrival is guaranteed, while CAN has to incorporate transmission collision detection &amp; correction which incurs non deterministic time overhead. 

On the other hand, ARINC 429 only uses a parity bit for bit level error detection whilst CAN allows for various methods including CRC checks

Some of this data is obtained from an article on CAN bus in avionics magazine. 

Newer ARINC busses like 629 or 664 (AFDX) allow for higher speeds and bi-directional communication. These may eventually replace ARINC 429 (but never underestimate how slow progress in aviation is)