For those who are unfamiliar with the difference between the two:
Munchausens is where someone fabricates or fakes symptoms or illness for attention, for example showing up at A&E pretending you have abdominal pains so you can undergo tests and receive attention and other benefits. Munchausens by proxy simply means you are doing it to another person, for example a mother feigning that her child has cancer by under feeding him, shaving his head, etc.
Health anxiety is an anxiety disorder where the person is preoccupied with worries about getting ill; the difference is, they are genuinely worried that they might be. They misinterpret all kinds of benign physical symptoms such as a headache as potentially meaning something catastrophic like a brain tumour. They are genuinely afraid they may have something wrong with them, and spend a lot of time trying to reassure themselves with doctor's visits, online research, physical monitoring (taking their own pulse for example). Due to being to anxious, sufferers often bring on anxiety symptoms such as chest pain or breathlessness which they then interpret as being signs of something dangerous like a heart attack, increasing their anxiety and intensifying the symptom in a nasty vicious cycle.
If someone presents to A&E with stomach pain, whatever its cause, and are afraid it might be something serious like appendicitis, and this is part of a wider pattern for them over time of seeking a lot of medical attention for minor ailments or sensations, along with the reassurance seeking from friends and family and the internet and so forth, it may indicate health anxiety.
If they present, knowing they don't have any pain, and fake it so they can be seen by a doctor, maybe given medication, given lots of time and attention from family, while knowing they have nothing wrong with them deep down, that may be munchausens.
Munchausens is more rare than health anxiety but still needs to be kept in mind when dealing with someone with frequent presentations to medical services with no evidence of anything wrong. I knew a chap recently who went to A&E saying he had end stage lung cancer and was under chemo in another part of the country and couldn't breathe, he explained he had multiple tumours and months to live.
The hospital gave him a scan and found he didn't have any cancer at all, perfectly healthy. They said this to him and his response was a 'oh, that's good news, thanks' and walking away. That is potentially munchausens.
If he'd arrived with actual mild breathlessness and mild chest pain terrified it might be cancer with no reason to think so, they may be health anxiety. If it's part of a wider pattern of him doing this repeatedly with different symptoms.
Hope that clears it up a bit!
It's worth noting that people faking illness for other practical gains like time off work or drugs aren't classed as munchausens, more malingerers. Munchausens is more about the person receiving attention and care rather then practical benefits from being sick.
Munchausens and health anxiety are both real illnesses. Ironically, the person who is faking illness for attention or terrified they have stomach cancer based on a bit of a stomach pain both are suffering from an illness, a mental illness rather than the physical one they are fabricating or irrationally afraid of.