You can just change date format to yyyy/MM/dd and all seems work
// Code goes here
angular.module("app",[]).controller('ctrl',function($scope){
  $scope.startDate = '/Date(1441045800000)/';
  $scope.mfstDate = new Date(2015,8,1);
})
<script src="https://ajax.googleapis.com/ajax/libs/angularjs/1.2.23/angular.min.js"></script>
<div ng-app="app" ng-controller="ctrl">
    <h1>Hello Plunker!</h1>
    {{startDate.slice(6,-2) | date:'dd/MM/yyyy'}}
    <div ng-if="(mfstDate| date:'yyyy/MM/dd')>(startDate.slice(6,-2) | date:'yyyy/MM/dd')" style="background-color:#E25D5D">{{startDate.slice(6,-2) | date:'dd/MM/yyyy'}} less {{mfstDate| date:'dd/MM/yyyy'}}</div>
    <div ng-if="(mfstDate| date:'yyyy/MM/dd')<(startDate.slice(6,-2) | date:'yyyy/MM/dd')" style="background-color:#E25D5D">{{startDate.slice(6,-2) | date:'dd/MM/yyyy'}} greater {{mfstDate| date:'dd/MM/yyyy'}}</div>
  </div>
 
 
UPDATE i think you need a bit fix your getFirst_Last_DateOfMonth function, you not need manually format string, because angular have a good filter date that you already yse inside view.
So you can return just date instead string,
function getFirst_Last_DateOfMonth(F_L) {
    //here F=First Date of Current month
    //     L=Last Date of Current Month
    var today = new Date();
    var dd = today.getDate();
    var mm = today.getMonth();
    var yyyy = today.getFullYear();
    var firstLastDay = F_L == 'F'? new Date(yyyy,mm,1) : new Date(yyyy, mm + 1, 0);
    return firstLastDay; //return Date object and in view use date filter when need comaring
    //return $filter('date')(firstLastDay,'yyyy/MM/dd'); //uncomment if want return formatted string
}
See snippet below
// Code goes here
angular.module("app", []).controller('ctrl', function($scope, $filter) {
  function getFirst_Last_DateOfMonth(F_L, formatString) {
    //here F=First Date of Current month
    //     L=Last Date of Current Month
    var today = new Date();
    var dd = today.getDate();
    var mm = today.getMonth();
    var yyyy = today.getFullYear();
    var firstLastDay = F_L == 'F' ? new Date(yyyy, mm, 1) : new Date(yyyy, mm + 1, 0);
    return !formatString ? firstLastDay //return Date object and in view use date filter when need comaring
      : $filter('date')(firstLastDay, 'yyyy/MM/dd'); //uncomment if want return formatted string
  }
  $scope.startDate = '/Date(1441045800000)/';
  $scope.mfstDate = getFirst_Last_DateOfMonth('F');
  $scope.mfstDateString = getFirst_Last_DateOfMonth('F', true);
})
<script src="https://ajax.googleapis.com/ajax/libs/angularjs/1.2.23/angular.min.js"></script>
<div ng-app="app" ng-controller="ctrl">
  <h1>Hello Plunker!</h1>
  {{startDate.slice(6,-2) | date:'dd/MM/yyyy'}}
  <div>
    mfstDate is date!
    <div ng-if="(mfstDate| date:'yyyy/MM/dd')>(startDate.slice(6,-2) | date:'yyyy/MM/dd')" style="background-color:#E25D5D">{{startDate.slice(6,-2) | date:'dd/MM/yyyy'}} less {{mfstDate| date:'dd/MM/yyyy'}}</div>
    <div ng-if="(mfstDate| date:'yyyy/MM/dd')<(startDate.slice(6,-2) | date:'yyyy/MM/dd')" style="background-color:#E25D5D">{{startDate.slice(6,-2) | date:'dd/MM/yyyy'}} greater {{mfstDate| date:'dd/MM/yyyy'}}</div>
  </div>
  <div>
    mfstDate is formatted string!
    <div ng-if="mfstDateString>(startDate.slice(6,-2) | date:'yyyy/MM/dd')" style="background-color:#E25D5D">{{startDate.slice(6,-2) | date:'dd/MM/yyyy'}} less {{mfstDateString}}</div>
    <div ng-if="mfstDateString<(startDate.slice(6,-2) | date:'yyyy/MM/dd')" style="background-color:#B8BCEF">{{startDate.slice(6,-2) | date:'dd/MM/yyyy'}} greater {{mfstDateString}}</div>
  </div>
</div>