In July and August the two are close: both are dry and sunny, with highs of about 28°C on Mykonos and 31°C on Rhodes, and a sea of 24-25°C against 27-28°C. The difference shows in the shoulder months. In September Rhodes' sea is still 27°C, in October it is 24°C, while Mykonos has cooled to 23°C and 21°C. If you plan a late trip, Rhodes is the island where you can still swim comfortably.
Both islands are windy in summer. The meltemi is the strong, dry north wind of the Aegean that blows from about mid-May to mid-September, strongest in the afternoon, and it can stop small boats. It matters more on Mykonos, where the north coast is exposed and boats can occasionally be cancelled in strong wind, though only on a few days a summer. On Rhodes the wind comes mostly from the west, so the east coast stays calm while the west is for surfers.
For Mykonos the sweet spots are mid-May to June and September to mid-October, when rooms are cheaper and the island is less packed. In March and April many restaurants and shops are still shut. Rhodes' international flights are seasonal too, so in winter you'd fly via Athens.
Neither island has a special safety issue, but the summer wildfire season applies to all of Greece. The UK Foreign Office rates the risk as high from April to October, and tells travelers to keep emergency alerts on, to call 112 if they see a fire, and to follow local emergency guidance.
Rhodes had a major fire in July 2023. An estimated 19,000 people were evacuated by land and sea, and the Greek authorities and media called it the largest wildfire evacuation the country had seen. The tourism ministry lifted the state of emergency on 2 August. Villages near Lindos were among those evacuated. It's a reason to read the local news before a late-summer trip, not a reason to avoid the island. See the UK travel advice before you go.