Difference 1:
you can call all elements matched by the JQuery object using trigger. 
//Example1  for trigger. All 3 button click events are fired when used trigger.
//Try Replacing trigger method with triggerHandler(). You will see only the first button element event handler will fire .
<button id = "button1">button1</button>
<button id = "button2">button2</button>
<button id = "button3">button3</button>
$("#button1").on("click", function(){
alert("button1 clicked");
});
$("#button2").on("click", function(){
alert("button2 clicked");
});
$("#button3").on("click", function(){
alert("button3 clicked");
});
//substitute trigger with triggerHandler to see the difference
$("#button1, #button2, #button3").trigger("click");
Difference 2: when using triggerHandler() for an element event, the native event will not be called for that element. trigger() will work fine.
//Example: 
//substitute trigger with triggerHandler to see the difference
 <button id = "button1">button1</button>
  <button id = "button2">button2</button>
$("#button1").on("click", function(){
 $("#button2").trigger('click');
});
$("#button3").on("click", function(){
var value = $("#button2").triggerHandler('click');
    alert('my value:'+ value)
});
$("#button2").on('click', function(){
alert("button2 clicked");
});
Difference 3: trigger() return Jquery object whereas triggerHandler() return the last handle value or If no handlers are triggered, it returns undefined
//Example3
<button id="button1">Button1</button>
<button id="button2">Button2</button>
<button id="button3">Button3</button>
$("#button1").on("click", function(){
var myValue = $("#button2").trigger('click');
    alert(myValue);
});
$("#button3").on("click", function(){
var value = $("#button2").triggerHandler('click');
    alert('my value:'+ value)
});
$("#button2").on('click', function(){
alert("button2 clicked");
    return true;
});
Other difference is
Events triggered with triggerHandler() do not bubble up the DOM hierarchy; if they are not handled by the target element directly, they do nothing.