Hi Lakshmi Narasimha

Action function
1. Difference is in case of Action function we invoke AJAX using Java script while in case of Action support we may directly
invoke method from controller.
2. Actionfunction is a function that can be called by any component via onclick, ondbclick, onblur event of that component.

Example code:
 

<apex:page controller="exampleCon">
<apex:form>
<!-- Define the JavaScript function sayHello-->
<apex:actionFunction name="sayHello" action="{!sayHello}" rerender="out"
status="myStatus"/>
</apex:form>
<apex:outputPanel id="out">
<apex:outputText value="Hello "/>
<apex:actionStatus startText="requesting..." id="myStatus">
<apex:facet name="stop">{!username}</apex:facet>
</apex:actionStatus>
</apex:outputPanel>
<!-- Call the sayHello JavaScript function using a script element-->
<script>window.setTimeout(sayHello,2000)</script>
<p><apex:outputText value="Clicked? {!state}" id="showstate" /></p>
<!-- Add the onclick event listener to a panel. When clicked, the panel triggers
the methodOneInJavascript actionFunction with a param -->
<apex:outputPanel onclick="methodOneInJavascript('Yes!')" styleClass="btn">
Click Me
</apex:outputPanel>
<apex:form>
<apex:actionFunction action="{!methodOne}" name="methodOneInJavascript"
rerender="showstate">
<apex:param name="firstParam" assignTo="{!state}" value="" />
</apex:actionFunction>
</apex:form>
</apex:page>

Controller class:

public class exampleCon {
String uname;
public String getUsername() {
return uname;
}
public PageReference sayHello() {
uname = UserInfo.getName();
return null;
}
public void setState(String n) {
state = n;
}
public String getState() {
return state;
}
public PageReference methodOne() {
return null;
}
private String state = 'no';
}

ActionSupport
1. other difference is Action function may be commonly used from different place on page while action support may only be used for particular single apex component.

2. ActionSupport is added to a particular component like button or outputtext and the event is controlled by the actionsupport not the component thus actionsupport help enhance the compononment itself by being part of the component itself.


Example Code:
 

<apex:page controller="exampleCon">
<apex:form>
<apex:outputpanel id="counter">
<apex:outputText value="Click Me!: {!count}"/>
<apex:actionSupport event="onclick"
action="{!incrementCounter}"
rerender="counter" status="counterStatus"/>
</apex:outputpanel>
<apex:actionStatus id="counterStatus"
startText=" (incrementing...)"
stopText=" (done)"/>
</apex:form>
</apex:page>

Controller Class:

public class exampleCon {
Integer count = 0;
public PageReference incrementCounter() {
count++;
return null;
}
public Integer getCount() {
return count;
}
}

Kindly mark it as solved if it helps you.

Best Regards,
Rakesh.T