The island is big enough that the beach you pick decides the holiday you get. Here is the honest difference between them.
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Phuket is fifty kilometres long, and the drive from the airport in the north to Rawai in the south takes well over an hour. That single fact decides more about your week than any hotel review: a cheap room in the wrong place costs you two hours a day in a minibus.

The west coast holds almost every beach worth the name, strung from Mai Khao in the north to Nai Harn in the south. The east coast is where the island works — piers, boatyards, the old town — and where almost nobody swims. Choose the west, then choose how loud you want it.
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Patong is the loud one and makes no apology for it. Everything is walkable — bars, tailors, massage shops, the beach itself — and it is the only part of Phuket where you genuinely do not need transport. If you want quiet, it is also the worst possible choice.
Kata and Karon sit just south and are the family compromise: wide sand, restaurants along the road, a manageable amount of noise. Karon is longer and calmer, Kata smaller and more social with a surf scene from May to October.

Nai Harn and Rawai in the far south are where long-stayers end up. Nai Harn has the better beach, Rawai the better seafood and a working fishing pier, and both feel like a town people actually live in rather than a resort strip.
The north-west — Bang Tao, Layan, Nai Yang, Mai Khao — is resort country: long, near-empty sand and hotels built to keep you inside them. It is the right answer for a honeymoon or a week with small children, and the wrong one if you want to wander out for dinner.

Kata or Karon. They have the swimming, the restaurants and enough transport to see the rest of the island, without Patong's volume.
Karon, Kata or the north-west resorts. Karon has the gentlest swimming and a long flat beach; the north-west has the space but you will need a car.
For a night or two, yes — the old town is the most interesting part of the island. But there is no beach, so most people combine it with a coastal stay.
Twenty minutes to Nai Yang and Mai Khao, forty-five to Patong, and an hour and a quarter to Nai Harn or Rawai in the south.
More Phuket and Andaman coast guides, plus the rest of Thailand's islands.