Elves did not have pointy ears, no.
Tolkien said (probably in the appendixes somewhere; I'm not renowned for using actual quotes...) something about the myths of elves flying, and having pointy ears, and being able to magik their way out of any situation, were all human-invented legends that had come about since the Fading of the Elves; since fact had been blurred. Elves, as you say, are genetically the same as humans; the reason they are more beautiful and graceful than humans is purely because of the sheer force of will of an Elf's fea.
Why don't the feas of men make their bodies immortal, and beautiful, like the Elves do? Maybe it's because Elves have a closer affinity to the Earth than Men. Men live for eighty years or so, and then dpart to another world; however Elves are bound to stay for hundreds of thousands, even Millions, of years. They love the works of creation, of nature, and, since they are going to be with them for millenia, they may as well take pleasure in them. They find greater joy in the world than Men, and accordingly, are more graceful and beautiful; more a part of the Earth itself than Men are.
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