What is difference between ABR and ASBR in OSPF network?

Well Abr stands for Area border router. Area 0 is a very special area in ospf which has the capability of sharing inter area routes. Abr is some thing in which at least one of its interface is connected to area 0. If incase you are not able to connect to area 0 physically you can do so logically by using the concept of ospf virtual link. 

Asbr stands for autonomous system boundary router when ever you perform redistribution by using commands like redistribute connected subnets (to redistribute the connected subnets to that router) then that router is called as asbr. 

                                                                            ABR And ASBR

Area Border Router ( ABR )  :  A Router is said to be area border router if it's minimum one interface is in OSPF AREA 0  and other interfaces can be in any area ( Area 1  -   Area 2^32-1 ).

To check whether a router is area border router or not Command  :   show ip protocols

ex. As you can see in above diagram Router R2 is ABR

R2(config)#do show ip protocols 
*** IP Routing is NSF aware ***

Routing Protocol is "ospf 2"
  Outgoing update filter list for all interfaces is not set
  Incoming update filter list for all interfaces is not set
  Router ID 2.2.2.2
  It is an area border router
  Number of areas in this router is 2. 2 normal 0 stub 0 nssa
  Maximum path: 4
  Routing for Networks:
    12.0.0.0 0.0.0.255 area 0
    23.0.0.0 0.0.0.255 area 1
  Routing Information Sources:
    Gateway         Distance      Last Update
    1.1.1.1              110      00:01:30
  Distance: (default is 110)

Autonomous System Boundary Router ( ASBR ) : A Router is said to be ASBR if it's one interface is in OSPF Domain and Other Interface in any Other Routing Protocol Domain ( RIP , EIGRP ) .  But we have to redistribute in order to make Router R3 as ASBR.

To Check whether a router is asbr command : show ip protocols

ex. In above example Router R3 is ASBR

R3(config)#router ospf 3   
R3(config-router)#redistribute eigrp 1 subnets 
R3(config-router)#ex
R3(config)#do show ip protocols
*** IP Routing is NSF aware ***

Routing Protocol is "ospf 3"
  Outgoing update filter list for all interfaces is not set
  Incoming update filter list for all interfaces is not set
  Router ID 3.3.3.3
  It is an autonomous system boundary router
 Redistributing External Routes from,
    eigrp 1, includes subnets in redistribution
  Number of areas in this router is 1. 1 normal 0 stub 0 nssa
  Maximum path: 4
  Routing for Networks:
    23.0.0.0 0.0.0.255 area 1
  Routing Information Sources:
    Gateway         Distance      Last Update
    2.2.2.2              110      00:09:52
  Distance: (default is 110)

ABR stand for (Area Border Router) while when we run OSPF (IGP) the area 0 or backbone are router is called ABR. Area 0 have a ability to share all routes in toplogy with others.

ASBR stand for (Autonomous system Boundary Router): when ae are apply router redistribution in OSPF the  one ASBR connect with ABR for comunicatrion ort data sharing.