Parsing string to date using javascript

Do not make things more complicated than necessary.
startdate_string is a string and String(startdate_string) make it a string, so

var startDate = new Date(String(startdate_string));

and

var startDate = new Date(startdate_string);

are the same thing.
when code fails, make sure of all what you can to narrow the search,
one solution is to try to print as much as possible to know where is the fail.

startdate_string = "2/21/2017 12:03 PM";
enddate_string = "2/21/2017 12:12 PM";
var startDate = new Date(startdate_string);
var endDate = new Date(enddate_string);
console.log(startDate);
console.log(endDate);
console.log(endDate-startDate);

Another solution is to use the debugger in your browser or use FireBug
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When you don't understand what your code is doing or why it does what it does, the answer is debugger.
Use the debugger to see what your code is doing. Just set a breakpoint and see your code performing, the debugger allow you to execute lines 1 by 1 and to inspect variables as it execute, it is an incredible learning tool.

Debugger - Wikipedia, the free encyclopedia[^]
Mastering Debugging in Visual Studio 2010 - A Beginner's Guide[^]

The debugger is here to show you what your code is doing and your task is to compare with what it should do.
There is no magic in the debugger, it don't find bugs, it just help you to. When the code don't do what is expected, you are close to a bug.

try this manual parsing

function convertToJSDate(datestr)
        {
           
            var parts = datestr.split(' ');
            var date = parts[0], time = parts[1], ampm = parts[2];
            var datepart = date.split('/');
            var timepart = time.split(':');
            var hours = timepart[0];
            var minutes = timepart[1];
            if (ampm == "PM" && hours < 12) hours = hours + 12;
            if (ampm == "AM" && hours == 12) hours = hours - 12; 
            
            var fianl = new Date(datepart[2], datepart[0] - 1, datepart[1], hours, minutes, 0, 0);
            return fianl;
        }
        convertToJSDate('2/21/2017 12:03 PM'); //Tue Feb 21 2017 12:03:00 GMT+0530 (India Standard Time)

demo : JSFiddle [^]

Never do that. You cannot reliably work on Date as string. Instead, use JavaScript Dates[^]. Refer to my previous answer[^].