The number of cases per thousand population aren't greatly different, but the number of deaths per thousand population are very different. Why is that? I cannot thoroughly explain it. Part is probably that Switzerland's over age 65 population is somewhat smaller, 18.4% versus Italy's 23%. However, this might be offset by Switzerland's higher percent of smokers. I have some suggestive ideas. One big difference is per capita health care spending. Switzerland is second highest in the world -- the USA being #1 -- and Italy below average among OECD countries. Switzerland's per capita spending was $7,719 in 2016; Italy's only $3,391 (Link). Switzerland had more hospital beds per 1,000 population in 2017 -- 4.53 versus Italy's 3.18 (link).
Public and compulsory spending on healthcare as a percent of the total in Switzerland is somewhat lower than Italy. I suspect the compulsory part for Switzerland is much higher, because the people are required to buy insurance with the government subsidizing the purchase. This page doesn't mention anything like Medicare. The compulsory health insurance is provided by private insurers. Unlike the USA, there appears to be no employer-provided health insurance.
Italy appears to have a Medicare-like system, but for all ages. In other words, it's Medicare for All. It's called the National Health Service. "In 2019, Italy's healthcare system was regarded, by World Health Organization's ranking, as the 2nd best in the world after France." Link. The WHO ranked Switzerland #20. Isn't that ironic? Yet Bernie Sanders wants the USA to be a lot more like Italy!
Edit 3/25/2020: Markets vs socialism: South Korea, Italy, COVID-19 Hat tip to dream_weaver.
Addenda 3/26/2020: Somebody replied to the above with the following point. There is no treatment for coronavirus, so I doubt that the healthcare system is what makes it different in this case.
I replied as follows. It seems more accurate to say there is no good for coronavirus. This article is about treatment. Some patients have been given drugs experimentally, which has helped some, and many are recovering from the virus. Aren't ventilators, which are often in news stories such as this one, a treatment? If a patient with COVID-19 has great difficulty breathing, can be put in a hospital on a ventilator, and survives, that strikes me as better than no hospital and no ventilator. It sounds plausible to me that the patient is better off in Switzerland than Italy or Spain.