HI Anita,

Whenever Save a record in salesforce instance.Salesforce fires a DML statement to its database either insert or update, internally.

1. All the code written in the "before update" triggers, executes BEFORE that DML is committed to Salesforce Database.
2. The code is written in after trigger executes AFTER the commit is made.

Hence if you are trying to update any value in the record, on which the trigger is fired, you need not write an update call, 
specifically.

example, if a before trigger is written on Contact and you want to change a value of one of the fields it would be
 

for(Contact eachContact: Trigger.new)
{
    if(eachContact.value__c='Temp')
    eachContact.Value__c = 'New value';
}

but if you will write this on after update you can't update the same field because data is already committed to database
so value will be read-only.

After update trigger generally works when you want to update any other object. 
for example, if you want to create a new task on Contact insertion or update you can write logic for that.

In before insert context your Trigger.NewMap always be null because in before context records is not submitted to database so Id is not generated that's why in before insert we don't use  Trigger.NewMap

But in After insert Id is generated so We can use Trigger.NewMap

In case of before and After update since id is already generated in Insert event.

SO we can use Trigger.NewMap in before and After update


Hope It will help you to understand main differences,

Thanks,
David(1040)