General case
This and many similar tasks were traditionally solved by reinventing the same solution over and over again, leading to problems during the development like you experienced here, and leaving a maintainability nightmare for the future.
Fortunately there is a small library available for exactly cases like this: Moment.js.
Counting days
To display how many days have passed since a given date:
var days = moment().diff("2015-06-02", "days");
It handles rounding for you, it handles daylight saving boundaries, it handles time zones, it handles date format parsing for you etc. - there is really no need to reinvent the wheel and write the same code many, many times.
See: DEMO.
But it gets even better than that:
More intelligent output
You can use this instead:
var ago = moment.duration(moment().diff("2015-06-05")).humanize() + " ago";
and you will automatically get values like:
"15 days ago"
"13 hours ago"
"5 years ago"
that would be ready to put in your countedBannedDays.innerHTML
giving your users more informative values. See: DEMO.
Your example
In your example, as long as the value of your banSelectedDate
contains a valid date, then your code - with the DOM handling code removed because that would stay the same:
var banDate = $find("<%= rdtpBan.ClientID %>");
var TodayDate = new Date();
var today = new Date();
var todayFormat = new Date(today.getMonth() + 1 + "/" + today.getDate() + "/" + today.getFullYear());
var banSelectedDate = banDate.get_selectedDate();
var one_day = 1000 * 60 * 60 * 24;
var countedDays = (Math.ceil(todayFormat - banSelectedDate) / one_day);
can be simplified to just:
var banDate = $find("<%= rdtpBan.ClientID %>");
var banSelectedDate = banDate.get_selectedDate();
var countedDays = moment().diff(banSelectedDate, "days");
with the relevant days counting logic being just:
var countedDays = moment().diff(banSelectedDate, "days");
It's obvious what's going on, you don't have to handle weird edge cases, you avoid exposing internal data representations leading to leaky abstractions, and there is really not much room to make any mistake.
Other options
There are also other libraries:
See more info in this answer.
With good libraries available to handle such common tasks we don't have to write complicated code with many potential problems like the ones you had here. Code reuse not only allows you to avoid potential problems but it makes your own code much more maintainable.