There are some basic differences between the two systems:
Sayan (Tropical) and Niryan (Sidereal)
Systems: In Indian Vedic system a year is completed when earth makes one revolution of sun (Sidereal year, 365 days 6 hours 9 minutes and 9.5 seconds). So if you observe the sun against a star, you would find it at the same star after a year).
Western system follows tropical (seasonal) year, and completion of year depends upon the equinoxes (the day on which day and night hours are equal at the equator) and a year is equal to 365 days 5 hours 48 minutes and 46 seconds.
That means a tropical year completes about 20 minutes before sidereal year. This phenomenon occurs due to earth’s special motion, and is called precession of equinoxes. Due to this reason the star which you have seen at mid night a year (tropical) before would be seen shifted by 50.26 seconds in latitudes (angular measurement: 1 circle = 360 degrees, 1 degree = 60 minutes and 1 minute = 60 seconds). Considering present difference 23°47’26” on 1st of January, 1995), the position of stars as observed in the two systems were the same in year 285 AD. This difference is called Ayanamsha in Indian system and the system having such difference (Sa+ayan) is called Sayan and that without (Ni+ayan) Niryan. A full precession cycle completes in about 25800 years. The annual motion of precession of equinoxes is also not a constant and varies gradually in cyclical manner from 50.25 to 50.30 seconds).
Similarly, stating point of the 12 zodiac in Western system is the spring equinox (March 21 when the sun is at the star zeta Pisces/ Piscium (Struve 100), though this position varies with time and another star will take this position. In Indian system, it is on April 14 (the Sun will always be at the star Beta Arietis (Sharatan) or Gamma Arietis (Mesrathim) (The full moon will be at 180 degrees opposit in that month at star Spica or Chitra and so the first Hindu month is Chaitra)
So, the relative position of planes remain the same in both the systems, the planetary position will be found changed slightly.
Indian system has another feature called Naksatras, which are stars or group of stars. Total 27 in numbers, they represent daily motion of moon which takes about 27 days to revolve around the earth. So in fact each of the 12 zodiacs is subdivided into nakshtras. So the astrological calculations become more complicated, they offer more permutations and combinations.
Rahu and Ketu:
Hindu system also have two opposite planets Rahu (North lunar node) and Ketu (South lunar node) The planets must be placed with sun and moon when an eclipse occurs. (To know more about Rahu & Ketu, please read other articles .in this website).
Uranus and Neptune:
Indian astrology do not take into account planets which can not be visualised through naked eyes, as their overall effect on humans become minimal due to their distance or size.
Sun Sign or Moon sign:
Indian system gives more importance to the moon sign of the persons rather than sun sign. Moon sign changes every 2 ½ days.
This is the most important feature of Hindu system. The Zodiac at the eastern horizon at the time of birth is called lagna. Its significance is considered more than any planet. (Lagna changes every 2 hours.
Dasha or Period or spell:
This is another feature of Hindu astrology which requires some calculations. It is believed that the human remains under the spell of different planets in a cycle of 120 years (many other cycles are also in vogue which are applied in different cases). The method is used for future prediction of events with precise timings.
Yoga (or Combination of planets and should be distinguished from Yoga exercises though the root of the word is the same). Learned sages have made list of thousands of permutations and combination of inter related planetary positions which cast their effect in a specific way or combination which if occur in a birth chart gives a certain result. This is a very useful way of obtaining precise prediction avoiding reinvention of wheel by every new astrologer.
There are many other major important differences which a scholar knows but I have not detailed them out to keep this article simple (suggestions and comments welcomed).