I'm in Australia. Not European but still interesting to compare.
There is a public and private system here, like the UK, but many people pay for private insurance, so they can be seen quicker, by a specialist they choose, often in a swankier hospital with fancy meals. You still often end up paying a fee on top of your insurance though when you access private care, because the insurance companies don't usually cover the full cost.
Anyway. I use the public system here. Some notable differences. Contraception is not free, and the initial upfront cost of long acting reversible contraception puts people off using it, rates are much lower than in Europe.
Ambulances are not covered by the government. If you need to be taken to hospital in one, you will be out of pocket. A life threatening emergency call out can cost you up to $1000 without insurance.
Physio, speech therapy, OT, podiatry and psychology services are not covered by the government. You need private insurance to access them.
Prescriptions are not free (like Scotland). Some are subsidised, but I've had hyperemesis this pregnancy, and the cost of antisickness medication has been eye-watering. I've tried to go without it numerous times because of cost and can't manage to.
GP care is a private system. It is down to the GPs discretion whether they accept a lower fee paid fully by the government for the consultation, thereby giving you it for free, but you don't find this out until after the consultation when you go to pay, so it's pot luck. This usually happens for students, kids and the elderly. If you are charged, you have to faff about getting half back as a government rebate, and paying the rest, usually about £25-50, depending on the practice. This cost has put me off going to the GP when I've considered it before.
There is a more consumer attitude toward healthcare. Like a pp said, tests and scans are sometimes done because patients expect them, rather than them actually offering useful information for your condition and being evidence-based.
Because health is a lucrative and expensive industry, alternative medicine is very popular, and pharmacies here have full aisles and large sections of various vitamins, supplements, natural treatments and homeopathy. Chiropractors, homeopaths and naturopaths are also very popular and used widely.
I've worked in the NHS and dear God, it might not be perfect in many ways, but it has a lot going for it.