Jeff is correct.

Avoid using apex whenever and wherever you can use standard functionality. 

For avoiding the recursion, There are two options:

1. Compare old and new values in entry criteria.

2. Controlling Recursive Triggers - This is the better way.

I tried to modify your trigger to check for old and new status value as entery criteria.

If this doesn't help, go for option#2

trigger TaskSendEmail on Task (after update)
{
// Don't forget this- all triggers in SF are bulk triggers and so
    // they can fire on multiple objects. So you need to process objects
    // in a FOR loop.
    Set<Id> CBIds = new Set<Id>();
   
    for(Task tsk: Trigger.New)
    {
    	if(tsk.RecordTypeId == '012Z00000004WFV' && tsk.Status == 'Completed' 
		&& Trigger.oldMap.get(tsk.Id).status != tsk.Status)
        {
        	CBIds.add(tsk.CreatedById);
        }
    }
       
   
    // Build a map of all users who created the tasks.
    Map<Id, User> userMap = new Map<Id,User>([select Name, Email from User where Id in :CBIds]);// Creating map for users who satisfied the conditions
    for(Task tsk : Trigger.New)
    {
        if(tsk.RecordTypeId == '012Z00000004WFV' && tsk.Status == 'Completed'
		&& Trigger.oldMap.get(tsk.Id).status != tsk.Status)
        {
        	User theUser = userMap.get(tsk.CreatedById);  // you will not have the user records for all the tasks createdbyIds
        
	        Messaging.SingleEmailMessage mail = new Messaging.SingleEmailMessage();
	        String[] toAddresses = new String[] {theUser.Email};
	        mail.setToAddresses(toAddresses);    // Set the TO addresses
	        mail.setSubject('The following Task has been Closed');    // Set the subject
	        // Next, create a string template. Specify {0}, {1} etc. in place of actual values.
	        // You can replace these values with a call to String.Format.
	        String template = 'Hello {0}, \nYour task has been modified. Here are the details - \n\n';
	        template+= 'Subject - {1}\n';
	        template+= 'Due Date - {2}\n';
	        String duedate = '';
	        
	        if (tsk.ActivityDate==null)
	            duedate = '';
	        else
	            duedate = tsk.ActivityDate.format();
	            
	        List<String> args = new List<String>();
	        args.add(theUser.Name);
	        args.add(tsk.Subject);
	        args.add(duedate);
	        
	       
	        // Here's the String.format() call.
	        String formattedHtml = String.format(template, args);
	       
	        mail.setPlainTextBody(formattedHtml);
	        Messaging.SendEmail(new Messaging.SingleEmailMessage[] {mail});
	    }
    }
}

On a side note: Never hardcode recordtypes in trigger, Either query them or fetch them from Schema using describe calls.

You cannot create an Email Alert fired by WF rule on Task object.