Pakistan observes its Independence Day on August 14, though it achieved freedom on the same day as India. The Independence of India Act is quite clear which states that on 15 August 1947 two independent Dominions shall be set up in India, to be known respectively as India and Pakistan. In fact, the first postage stamps that Pakistan printed have on them “15 August 1947” as the date of independence.
In Pakistan, however, this date was changed to August 14 in 1948 and I see the following possibilities -
1. Mountbatten delivered the King’s message of independence in Karachi on August 14, 1947 which is one day prior to the 15 Aug hence Pakistan chosen 14 Aug as its independence day.
2. August 14, 1948, was extremely holy in Islam (it was the 27th day of Ramzan) and Pakistan has chosen to be a Islamic country in place of being secular.
3. Or maybe, Pakistan just wanted to be a day ahead of India.
Whatever the reason, it lead to an incongruous situation where twins ended up with different birth dates.