Is it of any importance to know the time of day?

Time is the astrologer's god. However, few people know their real birth time. You are counting on some intern to get it right, if you are lucky. What if the intern forgot about about a time change and did not set her watch? Now you better have a GPS in the delivery room. Some countries do not even record the time. In the British Commonwealth nations, time is not required just day. Except for twins, first out is the legal heir.

Think about it. Up to modern times birth records were spotty at best. The most people did not have the disposable income to waste of astrologers. The rich were the few that did and that is how astrologers made a living. So for most of astrology's past existence maybe one percent of the population even bothered. It's simply not needed.

The time of birth is very important to know the degree and the rising sign (ascendant) and assigning planets and the houses. Without this the full horoscope cannot be cast.

Only after the full horoscope is casted, we can say with some accuracy the personality and character of the person.

Astrology is not a science. It is also anti-religion. It is a superstition and has grown up due to human weaknesses. I personally have studied astrology thoroughly and I believe in God. I tell you with conviction that it is not a science at all and is also anti-religion. Have you ever seen any of the religious teachers practising astrology ? Sankaracharyya, Ramanuja, Madhabacharya, Chaitnya Mahaprobhu, or any other religious men never practised it. Goutama the Buddha prohibited his disciples from studying and practising it.

time of birth is important in finding your rising sign.

Good question.

You see Sun stays in one sign (aries, taurus etc) for about a month. So, it wouldn't matter if you were born on 10th or 11th or 12th, because esentially Sun stays in that same sign for those 30 odd days. You will become a Virgo or something else only when Sun moves to a different sign. In an year, Sun moves at least once in every sign, that completes a zodiac.

Let's talk about other part of your question... if you know a little bit about horoscopes, you'll agree the 10th house of the horoscope is the place highest in sky. When Sun is at it's peak during the day, i.e. around noon 12PM, it's at the highest point in sky. Still with me? So, if you were born, let's say around 12 PM, on 08/10 (when Sun is in Leo), you should have Sun falling in 10th house of your horoscope. Now, since Sun is in Leo at that time, you should have Leo as well in 10th house, right? So, you can say you have Sun in Leo in 10th house. And then you can calculate other houses accordingly. Since Leo is in 10th, Virgo will be in 11th, Libra in 12th, Scorpio in 1st and so on. With me?

Do you know Ascendant (1st house) is the time when a sign rises on the eastern horizon? So instead of 12PM, if you were born around 10AM you will have Sun somwhere in 11th house, and if you were born around 8am on that same day, you'll find sun somwhere in 12th house of your horoscope. Reduce 2 more hours and you're likely to have Sun probably in your ascendant or near by. But on that day, irrespective of time, Sun continues to stay in that same sign. So, for 8am birth, you could expect Sun in Leo in 12th house instead of 10th house. And then you can calculate all other signs accordingly. So, after nearly two hours, rising signs change on eastern horozon. 12 sign * 2 hours = 24 hours = 12 houses of horoscope. Getting me? So if you see for all people born around 9AM on any day in the year, you'll see Sun lingering somewhere in 12th house, irrespective of sign.

Taking the example above, if you were born on 12PM, you'll have Sun in Leo in 10th house. Per Astrology's first rule, Any planet in it's own sign does good for the house which it owns in a horoscope. So, for this subject with Sun in leo in 10th, s/he can expect very good results in matters related to Profession (10th house matter). But if this native was born on lets say 12AM, Sun would be directly opposite, somewhere in 4th house. Now, instead of doing good for the profession, Sun will do good for all matters related to 4th house such as Mother, Property, Education, Vehicles because Sun being in Leo in 4th house owns it.

You see, in about a matter of 2 hours on each day, your entire destiny can change, and it always does.