What extra benefits are there in getting travel insurance if I have a European Health Insurance Card (EHIC)?

Often travel insurance is not just medical travel insurance but will cover general travel costs. Eg your flight is delayed and you want to book another relight / need to stay in a hotel, your hotel cancels your booking at the last minute and you need to book another hotel to stay in, your bags get lost in transit and you need to buy some new clothes etc. These are sort of things that an EHIC will not cover.

Additionally, travel medical insurance will probably allow you to see a private medical professional. The EHIC is just proof that you are entitled to the country 's public health system. This is not always the same as private medical cover. Eg if you have a major accident (break your arm) you might be able to go to a private hospital / clinic and be seen quicker, rather than waiting for hours in A&E, the travel medical might cover it. Travel medical insurance might cover repatriation, ie if you die, it will pay for the costs of sending your body back to your home country. I don't know if a EHIC will pay for that.

The other two answers at this point are excellent, but I just wanted to expound on one point: Travel insurance is almost always useful if/when a serious illness during your travel would require urgent transport to a faraway medical facility. It's not just limited to more extreme conditions like mountain rescues; for example, say you have a life threatening condition (e.g., appendicitis, heart attack, etc.) arise on a cruise ship in the middle of the Mediterranean. A ship carrying thousands of passengers will not make a detour just because one has a life threatening condition, and even if it did it might be tens of hours from the closest port at full speed. Therefore, such conditions necessitate a helicopter medivac to the closest hospital, which can cost tens if not hundreds of thousands of Euros. Whether or not your domestic insurance provider will cover those costs will vary from provider to provider, but here in the US even the most premium insurance plans do not cover such expenses.

As an anecdote, several years ago I attended a conference in India in which an Italian colleague was involved in a rickshaw crash, sustaining moderate injuries including at least one broken bone. I do not know all of the details of his case, but I do know that he did not have travel insurance and ended up having to depart early and fly back to Italy to receive treatment. Granted, I do not know if his decision to leave was due to a lack of coverage for the medical costs versus a preference to be treated by Italian doctors, but I got the impression that it was the former.

As I understand it the big one is covering expenses that are a consequence of a serious medical emergency (either injury or otherwise) but aren't medical expenses per-se.

Lets consider a scenario. You are a long way from home, suddenly you get hit by a bus (or whatever). You survive but with serious injuries including multiple broken bones. You get taken to hospital in an ambulance where they stablise you. Your significant other stays with you in the foreign country until you are stable enough to be transported home. However your broken bones are still bad enough that special provisions need to be made on your flight home.

The EHIC will cover your ambulance ride and stay in hospital under the same terms the locals will get (so either free or highly subsidised).

What the EHIC won't cover is the cost of your significant other staying in the area or the cost of your transport home. The latter in particular can be extremely expensive. Private travel insurance is almost certain to cover the cost of getting you home and may also cover other related expenses (read your policy carefully)

Other things like covering theft, companies you pre-purchased services from going bust etc are nice to have but less essential.