[SalesForce] Trigger on User Object when creating new user

As @sfdcfox noted, your immediate issue was ! = should be !=. However, you have several other issues, ranging from additional compile-time errors to inefficient code to bad practices which really should be addressed.

Here's what I see:

• You don't need to pass trigger.oldMap to an insert handler; it will always be null (there are no 'old' values for an insert). See Trigger Context Variables in the Apex Programmers guide for more. If you plan to expand this trigger handler for updates, you can add it back for testing if the user's email has changed.
• You don't need to test mapNewUser for null or empty; the insert trigger won't be called if there are no new User records being inserted. If you expand the trigger handler for updates, you will need to check if oldMap is null.
• Critical: you are running SOQL queries inside a for loop. This is a quick way to hit the SOQL query limit, and a very bad practice. I recommend reading all of the Apex docs on triggers, but especially Common Bulk Trigger Idioms.
• You declare lstAppAuthContoupdate and lstAuthContacttodelete inside a block, but try to use them outside the block. This will result in additional compile time errors.
• You are catching errors in your trigger handler, and then doing nothing but logging them. This might be okay for your use case; just note that these caught-and-suppressed errors will not prevent the new users from being inserted, and the only record of an issue will be in a logfile. If new user creation should be halted for these errors, either remove the error handler (letting the error propagate to the caller), or use [object].addError() in your handler to add errors to affected user objects. If you don't want to halt new user creation, consider emailing the errors to an Administrator.

Here's a reworked version of your code, using only one SOQL query and using maps to avoid the need for more queries. Note that you'll have to update your trigger to only pass trigger.newMap.

public with sharing class UserTriggerMethod {
    public static void updateApplicationEditor(Map<ID,User> mapNewUser) 
    {   

        // get a list of emails for the newly inserted users
        List<String> emails = new List<String>();
        for (User u : mapNewUser.values()) {
            emails.add(u.email);
        }

        //query all auth contacts (and associated App Auth Contacts) at once
        List<Auth_Contact__c> AuthContacts = [ 
            SELECT id, Email__c, 
                       ( SELECT Contact__c from App_Auth_Contacts__r) 
              FROM Auth_Contact__c 
             WHERE Email__c in :emails ];

        //create a lookup from email to AuthContact
        map<string, Auth_Contact__c> emailToAuthContactMap= new map<String, Auth_Contact__c>();
        for(Auth_Contact__c ac : AuthContacts) {
            emailToAuthContactMap.put(ac.Email__c, ac);
        }

        List<App_Auth_Contact__c> lstAppAuthContoupdate;
        List<Auth_Contact__c> lstAuthContacttodelete;
        for(User user : mapNewUser.values()) {

            if(emailToAuthContactMap.containsKey(user.email)) { 

                Auth_Contact__c AuthContact = emailToAuthContactMap.get(user.email);

                List<App_Auth_Contact__c> lstAppAuthCon = AuthContact.App_Auth_Contacts__r;              

                lstAppAuthContoupdate = new List<App_Auth_Contact__c>();
                lstAuthContacttodelete = new List<Auth_Contact__c>();

                for(App_Auth_Contact__c AppAuthCon: lstAppAuthCon)
                {
                    AppAuthCon.Contact__c = user.ContactId;
                    lstAppAuthContoupdate.add(AppAuthCon);
                }

                lstAuthContacttodelete.add(AuthContact);

            }
        }

        if(lstAppAuthContoupdate!=null && lstAppAuthContoupdate.size() > 0)
        {
            try
            {
                update lstAppAuthContoupdate;
            }
            catch(DMLException e)
            {
                system.debug(e.getMessage());
            }
        }

        if(!lstAuthContacttodelete.IsEmpty())
        {
            delete lstAuthContacttodelete;
        }
    }
}