Easiest to copy the answer from those nice puppies at the Rice Association....
Basmati Rice
A very long, slender grained aromatic rice grown mainly in the foothills of the Himalayas in India and Pakistan. Sometimes described as the 'Prince of Rice'. It has a fragrant flavour and aroma and is the rice used in Indian dishes. The grains are separate and fluffy when cooked. In Indian recipes it is often cooked with spices to enhance the grain's aromatic properties. Easy cook basmati and brown rice basmati are also available. Brown basmati rice has a higher fibre content and an even stronger aroma than basmati white.
Jasmine Rice (Thai Fragrant Rice)
Another aromatic rice, although its flavour is slightly less pronounced than basmati. It originates from Thailand. The length and slenderness of the grains suggest that they should remain separate on cooking but it differs from other long grain rices in that it has a soft and slightly sticky texture when cooked. Good with Chinese and South East Asian food.
Summary then - Jasmine is stickier when cooked. More than you want to know at:
http://www.riceassociation.org.uk
There are thousands of rice varieties - its really just grass seed - the red version from Camargue or the black wild rice is the most 'obviously' grass when you look at the shape of the grain.
And don't forget Chinese New Year on the 18th Feb - Year of the Piggy coming up!!!! Any excuse to get into the woks..