Although many people tend to use the words synonymously these days, careful users of the language maintain a distinction between the two. Do you “choose” your life partner, or do you “pick” one? Normally people “choose” their significant other. Choose suggests that you have opted for the individual after giving the matter some serious thought. You have weighed the pros and cons of an issue — in this case, marrying a person — and have arrived at a considered decision. It is voluntary.
Pick also means “selection”, but in this case, there may not be any serious thinking involved. The word suggests that you have opted for something in a very casual manner — there may not be any decision making or discrimination involved. For example, when you go to the supermarket and see several cans of the same product, you normally don’t take too much time thinking about which can to buy. Since the same company has made them, you “pick” the can that is nearest you. Most of us do not select our life partner in this casual manner.
Similarly, we “choose” a present for someone we really like and care about — we buy the present keeping in mind the person’s likes and dislikes. If we “pick” a present, what we are doing is buying something in a casual manner. We don’t keep in mind the individual’s likes and dislikes; it suggests that we buy the first thing we see in the shop.