I have never had the pleasure of casting horoscopes for twins. But the rule of astrology is first to know the birth time, to establish the beginning or ascendant on the chart. This changes every two hours. So depending on whether the Ascendant changed or the house, or planets on the cusps of the houses changed, it can alter how the whole astrology wheel is cast. There are 29 degrees in each astrology sign, so even though the second twin was born a few minutes later, some factors could have changed in that brief period. The Sun and Moon move fast, as does the sign ascending. So if they were in that late 29-degree point, the signs on the ascendant and on the houses could change in that short period.
We call the critical part of the horoscope a Trinity. If the second twin is born a few minutes later, or the moon or Sun changes zodiac signs during that period, this would account for differences between the twins. So one could have Aries rising, or the next would have Taurus rising. This will change what signs rule each of the 12 houses. But the 12 planets will be in the same signs, just at different parts or houses, and that would show up as different traits. But since the planets will be in the same signs, the twins will still have pretty similar charts. As I said, I've never been able to test this out!