This can change depending on who you ask.
Most people use psychotherapist and counsellor to refer to someone who delivers talking therapy. Those two are often used interchangeably. To become a registered therapist (or counsellor) you need to take a course which meets criteria as set out by a governing body. There are a few main ones, and if you fancy looking at them you have the BACP, UKCP, and the NCPS to name a few (UK based). I would recommend looking at them before you choose a therapist, they all have directories that you can search but they also have ethical codes and common questions and concerns. In the case of a counsellor or therapist it is important to find out which ethical organisation they are attached to, in case they are not qualified.
A psychiatrist is a doctor, and this means that they will normally use a mixed model, and be able to diagnose and prescribe medications. As a result in the UK the main way to access a psychiatrist is through the NHS or through a private healthcare provider.
There is some cross over between these roles and a clinical psychologist and a counselling psychologist, which are two more common mental health practitioners, and will often be doctors, but not medical doctors. They are also accessible within the NHS. These are protected titles in the UK so you can be sure that anyone using them is a legitimate option for treatment. (You can check on the medical register)
When you are deciding between a psychiatrist and therapist there may need to be a discussion about severity, for example, if you are currently struggling with maintaining contact with reality it may mean that you need to speak to a psychiatrist first. This would be considered a stabilisation stage. This means that the space that you're in will be more accessible to talking therapies.
If your problems also lie outside of what might be considered a mental disorder then a therapist is going to be able to help you when you may not be able to access a psychiatrist. This is things like relationship struggles, and behaviour patterns that aren't serving you.
To me this difference boils down to management of things that you would want to change and addressing things that stop you from living your life the way you want.
The important thing to know is that you don't have to be sure what you need. Anyone one of these should be able to redirect you if you would benefit from the expertise of the other. It is also fairly common that one person may need to talk to a number of the different fields at once, to make use of the different skill sets to make sure that there is best care. For example in the NHS a counsellor can refer to a Mental health team which will have a number of different specialists within them and would often triage or refer on to a more specialist service. A GP or mental health team also can refer to a counselling service to help a client.
When you are signposted to a different service some people can feel frustrated, espcially with long wait times. When we signpost as practitioners we do it because we want you to be able to access the help you need as quickly as possible. Sometimes it doesn't mean an end, for example, if I signpost and make a referral to a different service I try to carry on any sessions as long as the client can. I do this if there is work we can still do. My goal with this is to make sure that clients have support. This isn't possible all the time, but can help to make wait times feel more manageable.
(Don't be afraid to chase up wait lists, it can help to know where you are)
The main difference that you are likely to feel is the way that challenges are viewed, as a therapist I think of it more as individual challenges, but a more medicalised view may view the same things as symptoms. The way that this feels is very individual and there is no right answer.
That being said due to the far reaching consequences of mental health the challenges can be really varied such as impotence, hair loss and poor sleep to name a few. These challenges can be related to a variety of different mental health concerns, such as self worth and anxiety etc... but they may also be more medical. It is also possible that they are both. In cases like these it is important to also seek help from your GP, as much as mental health support can help alleviate these symptoms, there may be an underlying medical issue that needs to be addressed. A good therapist will be aware of this and be able to guide you, of when you need to speak to your GP to rule these things out. Often times it can be a good strategy to address the problem through as many different views as possible.
Long story short we are all a team doing our best to help you with your challenges.