Iodine is taken up and concentrated by the thyroid gland. Likewise, so is radioactive iodine (radioiodine). Hence, the administration of radioiodine makes it is possible to target and deliver radiation to the thyroid gland specifically, with essentially no exposure to other parts of the body.
There are two main reasons that someone might receive radioiodine - hyperthyroidism and thyroid cancer. The dose of radioiodine used for hyperthyroidism is much less than that for thyroid cancer.
For the relatively low doses used to treat hyperthyroidism, there are very few early side effects. Some people get a mild sore throat. There is also a risk of a worsening or precipitating the ocular manifestations (ophthalmopathy) of Graves Disease (if that was the cause of the hyperthyroid state in the first place). In the longer term, there is a high risk of developing hypothyroidism (about 20% within the first months and possibly 100% eventually). Note that this treatment does not require hospitalization.
There is no increase in cancer mortality after radioiodine treatment for hyperthyroidism, although thyroid cancer does occur more commonly in people who were treated with radioiodine than those who were not. Look here.
On the other hand, radioiodine treatment for thyroid cancer usually does require hospitalization. There is little doubt (IMO) about the efficacy of radioiodine treatment for thyroid cancer which makes acceptance of the side effects somewhat easier to bear, eg. sore throat (acutely), decreased salivation (longer term). Since “only” the thyroid gets the radiation, hair loss, nausea, and other “radiation toxicities” do not occur with this type of radiation therapy. Between the surgery and radioiodine treatment for thryoid cancer, hypothyroidism will definitely occur. But that’s almost irrelevelant since everyone with thyroid cancer will have to take thyroid hormone supplements anyway (to suppress any remaing thyroid cancer cells).