Pick the beach before the island, because the bases differ more than the two places do. In Phuket, Kata is the easy choice for families and first-timers: a one-kilometer beach, a village feel and a smaller sister beach, Kata Noi, over the hill. Karon is its longer, quieter neighbor, with hotels across the road from the sand, which suits walkers and couples. Kamala is smaller and calmer, a short ride from Patong's nightlife. Patong is where you stay to walk to the bars, shops and the widest choice of restaurants, and where you don't stay if you want sleep. The north (Mai Khao, Nai Yang, Nai Thon) is quiet and near the airport, but the other beaches are 20-60 minutes away, so you need a car or a lot of Grab rides. Lifeguard cover is reportedly thinner at Kamala than at Karon or Patong, so check the flags wherever you stay.
In Krabi, Ao Nang is the practical base: the boats to Railay, the Four Islands and Hong leave from here, the airport is about 40 minutes away and it has the most food and tours. It's also the most commercial beach in the province, and its sand is shallow and muddy at low tide. Railay has no cars and the best-looking beaches, and it's the pick for couples, climbers and anyone who wants a quiet evening, but it's boat-only, so luggage means wading to the boat, food costs more and the crossing is rough in the monsoon. Klong Muang, about 10 km from Ao Nang, is calmer and quieter, but you need a taxi or a rental for everything, and the Railay and Four Islands boats don't leave from there.
Koh Lanta is the other Krabi-province option: a long, lightly developed island with most of its accommodation in simple bungalows, popular with families, divers and people who want sunsets instead of nightlife. From Ao Nang it's about 2 hours by road plus car ferries, and you give up the easy boats to Railay and the Four Islands, so treat it as a different trip rather than a third option in this comparison.
Families are better served by Phuket: Kata and Karon put a long, flat beach next to hotels, shops and restaurants, the airport is closer, and the Old Town, Big Buddha and Wat Chalong give you something for a rainy afternoon. Krabi is easier in one way, since Ao Nang is compact and you can walk to dinner, but the best beach is a boat ride away and the Tiger Cave steps and the Railay viewpoint are not for small children or anyone unsteady. Couples have a closer call, and Krabi is no worse here: Railay or Klong Muang for scenery and quiet evenings, or Karon and Kamala if you'd rather have a flight without a connection.
Healthcare matters on a long trip. Thailand's private hospitals are good but ask for a payment guarantee before treatment, so travel insurance with medical cover is the thing to sort out, for either base. See the UK health advice.