The most obvious difference between the two is the climate. If you travel up in elevation, you are going to get into a colder climate. If you travel closer to the equator you are going to get into a warmer climate. If someone dislikes cold weather, he or she may have a hard time vacationing in the mountains. You will see many people in multiple layers of clothing. On the other hand, the beach offers a warm climate. The temperature at a beach is always hot, due mostly to its location near the equator. You will see many bikinis and lots of naked skin.
The cold climate of the mountaians offer an abundance of activities. Snow boarding, mountain biking, hiking, skiing and ice skating are a few. When people vacation in the mountains there evenings consist of sipping hot chocolate around a warm fire. If seclusion is what you seek then the mountains can satisfy that for you. However, the ocean and the warm climate of the beach offers swimming, volleyball, soccer, water biking, parasailing, deep sea fishing, snorkeling and many more. Most people go to the beach to lay out in the sun and tan. At most coastal sites, there are clubs and restraunts where people dance and party throughout the night. You can always expect popular beaches to be crowded especially during the summer time.
Out of all the places in the world I chose these as my favorite destinations. Whistler, Canada is popular in North America and has beautiful scenary and amazing lodging to accomodate you.