That commented out block can be made to work, with some tweaking. A problem arises because of the nature of your data, and the fact that the tutorial didn't take into account the way that object comparisons work. (Refer to JavaScript Date Object Comparison.)

Your first column consists of Date objects, which the == comparator will evaluate true only when both sides of the comparison are the same object. In the whole-row comparison, the dates are being coerced to strings by the .join() operation. We can get the same effect on a cell-by-cell basis, like this:

 if(row[0].toString() == newData[j][0].toString() && row[1] == newData[j][1]){
   duplicate = true;
 }

Alternatively, we can restrict the comparison to the first two columns by using the Array.slice() operation. This way, we don't need to know what types are being compared, because we'll still use .join() to form a string for the comparison:

if(row.slice(0,2).join() == newData[j].slice(0,2).join()){
  duplicate = true;
}

Sorting

Another issue you'll run into is your placement of range.sort(). It's dangerous to sort before removing duplicates, because you can't guarantee the order of newly-retrieved event data vs data for the same event that you already had which may have additional columns of information. As a result, the duplicate removal could erase your user-entered info. It's safer to leave sorting to be a final step, or include additional columns in the sort to guarantee order.

Another little issue: The use of .getRange("A3:F2000") has a side-effect of expanding your spreadsheet to 2000 lines; you could instead use .getRange("A3:F"), which will retrieve the maximum range without expanding it.

But I suggest you do the sorting entirely with javascript arrays, which will be much faster than using spreadsheet services. I'm assuming you have two lines of header information that you want to retain at the top of the sheet, since you were sorting from A3. Here's a way to do the sort:

// sort ascending by date - retain header lines
var headers = newData.slice(0,2);
var sorted = newData.slice(2).sort(sortFunction);
var newData = headers.concat(sorted);

Where sortFunction() is:

function sortFunction( a, b ) {
  // coerce dates to numbers and return comparison
  return ((+a[0]) - (+b[0]))
}

Final Script

Here's what you get, with the above changes.

function importEvents(){
  var calID = Browser.inputBox("Please enter your google Cal ID", Browser.Buttons.OK_CANCEL);
  var startdate = Browser.inputBox("Start date using 1/1/2013 format", Browser.Buttons.OK_CANCEL);
  var cal = CalendarApp.getCalendarById(calID);
  var events_sheet = SpreadsheetApp.getActiveSheet();
  var events = cal.getEvents(new Date(startdate), new Date("1/1/2014"));
  var lr = events_sheet.getLastRow();
  var eventarray = new Array();

  var i = 0; // edited
  for (i = 0; i < events.length; i++) {
    line = new Array();
    line.push(events[i].getStartTime());
    line.push(events[i].getTitle());

     //Potential more data that I am not getting at this time
     // line.push(events[i].getDescription());
     // line.push(events[i].getEndTime());

    eventarray.push(line);
  }

  // Append the retreived events to existing spreadsheet
  events_sheet.getRange("A"+(lr+1)+":B"+(lr+i)).setValues(eventarray);

  //remove duplicate rows
  var sheet = SpreadsheetApp.getActiveSheet();
  var data = sheet.getDataRange().getValues();
  var newData = new Array();
  for(i in data){
    var row = data[i];
    var duplicate = false;
    for(j in newData){

      if(row.slice(0,2).join() == newData[j].slice(0,2).join()){
        duplicate = true;
      }
    }
    if(!duplicate){
      newData.push(row);
    }
  }

  // sortFunction used to compare rows of data in our newData array
  function sortFunction( a, b ) {
    // coerce dates to numbers and return comparison
    return ((+a[0]) - (+b[0]))
  }

  // sort ascending by date - retain header lines
  var headers = newData.slice(0,2);
  var sorted = newData.slice(2).sort(sortFunction);
  var newData = headers.concat(sorted);

  // Clear the existing info and update with newData.
  sheet.clearContents();
  sheet.getRange(1, 1, newData.length, newData[0].length).setValues(newData);
}