In JavaScript, dates can be transformed to the number of milliseconds since the epoc by calling the getTime() method or just using the date in a numeric expression.
So to get the difference, just subtract the two dates.
To create a new date based on the difference, just pass the number of milliseconds in the constructor.
var oldBegin = ...
var oldEnd = ...
var newBegin = ...
var newEnd = new Date(newBegin + oldEnd - oldBegin);
This should just work
EDIT: Fixed bug pointed by @bdukes
EDIT:
For an explanation of the behavior, oldBegin, oldEnd, and newBegin are Date instances. Calling operators + and - will trigger Javascript auto casting and will automatically call the valueOf() prototype method of those objects. It happens that the valueOf() method is implemented in the Date object as a call to getTime().
So basically: date.getTime() === date.valueOf() === (0 + date) === (+date)