Both leads to the evolution by varing the gene frequency of population.The major difference is that the natural selection increases the trait's frequency more adaptive to environment  as compared to genetic drift , it rarely results in more adaptive species to our environment...
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The core difference between Natural Selection (NS) and Genetic (or allelic) Drift (GD) is cause. Both are methods of genetic change in a population however one happens randomly (GD) while the other is a direct response to and environmental challenge (NS). Now for the examples; 

Natural Selection as you probably know, is the mechanism by which a species (or genetic makeup of a breeding group) changes in response to an environmental challenge. For example, imagine a population of brown rabbits in a field. They are happy, breeding and being eaten occasionally by foxes until over time, the environment changes so that the field is covered in snow. 
Now, the brown rabbits stick out and the foxes have an easy time of hunting them. Consequently, the number of brown rabbits decreases dramatically and they are threatened with extinction. The genetic mutations always present in the population (and which increase with in-breeding in small populations) throw up random variations as always but now, some are more beneficial. For instance white rabbits which used to be caught and eaten quickly before the snow came, are now much better adapted. As such, they are more likely to survive, breed and pass on their white genetic make up and hence more white rabbits are born - they are naturally selected by the snow and the foxes; their environment. 

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