Difference Between : Routing action Vs Service Callout Vs Publish action in OSB

Routing action:

•  When you need to call a synchronous service then use it.
• In the message flow, regardless of whether routing takes place or not, the route node represents the transition from processing a request to processing a response.
• Always Last Node in Message Processing
• Only one Route node in a Proxy
• A Route Node indicates that Request Processing will end here and Response Processing will begin
• At the route node, the direction of the message flow is reversed.
• Routing action can be created inside a Route Node only

Service Callout :

• When you need to call a synchronous service and want to save req and resp variable for validation and enrichment.
• Service Callout is used in real time request-response scenarios (for sync response).
• If you have a use case where you need to call one or multiple real time request-response services (like sync web-services) then you may use Service Callout action in the message flow. 
• Being a synchronous call, the thread gets blocked until a response is returned from target service.
• Can have multiple Service Callout nodes in a Proxy service

Publish action :

• Publish action is used for Request only scenarios where you don't expect a response back
• The nature of Publish action (sync or async) will depend upon the target service you are invoking.
• If you are invoking an external service through a business service then Publish action with Quality of Service(QoS) as "Best Effort" (default) will work like fire and forget and thread won't get blocked (async call). 
• If you are invoking a local proxy service (proxy with transport protocol as "local") from another proxy using publish action then it would be a blocking call (synchronus call) and thread will get blocked untill the processing of local proxy finishes.

ref. - https://community.oracle.com/thread/1774545