The more grains, the heavier the bullet. In some instances, you may need a heavier bullet in order to insure better penetration and a quicker kill, such as when taking large or dangerous game like elk, moose, bear, etc. In other situations, you want a lighter bullet for a faster, flatter trajectory and/or when penetration isn't needed or wanted, such as when hunting smaller game animals like coyotes or varmints.
The weight of the bullet can also affect the bullet's ballistic coefficient, or BC. This is the measure of the bullet's ability to overcome air resistance in flight, and is usually calculated by dividing the mass with the bullet's diameter squared that it presents to the airflow divided by a dimensionless constant i that relates to the aerodynamics of its shape..
To put it simply, a heavier bullet is generally longer and more spear like, which will buck the wind better and fly flatter than a bullet that is shorter, lighter and closer to a ball or square shape. In addition, it will usually penetrate better.
However, a heavier bullet also means higher recoil.
The bullets BC and weight can also affect accuracy; if the rifle's barrel has too slow or too fast a twist rate to stabilize a bullet of that size properly, it will not group well and will generally be inaccurate, though this does not seem to be a hard and fast rule; some calibers seem to be more tolerant of over/understabalization than others, and it can very from rifle to rifle.