"Vegan" is a popular term in everyday language, while "plant-based" often appears in scientific literature. Is there any difference in meaning between the two terms?
The term vegan was intended by those who coined it to encompass lifestyles that respect non-human animals and refrain from any exploitation of them.
Although the vegan diet was defined early on it was as late as 1949 before Leslie J Cross pointed out that the society lacked a definition of veganism and he suggested “[t]he principle of the emancipation of animals from exploitation by man”.This is later clarified as “to seek an end to the use of animals by man for food, commodities, work, hunting, vivisection, and by all other uses involving exploitation of animal life by man”.
The language is decidedly dated, and I hope veganism can be defined more inclusively (in terms of gender and as rooted in multiple belief-systems) but I think it would be reasonable to say it is a term implying an ethical stance.
To say that someone is following a plant-based diet is to say they are eating no animal products (presumably without other lifestyle changes). There is no philosophical position or belief system implied behind their practice. This makes it useful for the purpose of reporting on research, where participants may have switched to a particular diet for the purpose of a study. It also includes people who don't eat animals but don't consider themselves vegan.
It is important to respect plants as much as animals. It is important to respect all species. So for this we have the phrase "ethical diet". It is important to have an ethical diet that respects all species of plants and all species of animals.  
An ethical diet consists of food and drink that are produced without hurting any plants or animals.
One definition of "vegan" is a diet that is ethical.
A vegan diet is known to be a lot safer and healthier than a non-vegan diet.
The website cancer.gov informs us that there is a high number of carcinogens in meat products. There are carcinogens present in meat products before the meat is cooked, and there are carcinogens that emerge after the meat is cooked. 
A vegan diet protects us from cancer and disease.
One of the most important facts of veganism is that we need to eat in order to stay healthy, and we need to be well-nourished in order to stay healthy. So it's super important to get all of our needed vitamins and minerals. We can learn about dietary sources of vitamins and minerals on the nutrition.gov website.
It is also super important to be hydrated. Drinking water is a form of hydration, and so is showering. Drinking water hydrates our cells and showering hydrates our cells. That water is really important for cellular metabolism and biological processes.
I am a vegan.
I have an ethical diet.
I try my best to have a diet that respects all plants, all animals, and all species of life.
 All plants, all animals, all organisms, all things, all species of life, are people. Every thing is a person. Everyone is a person. This has been known since the beginning of time. Every person has the right to be completely safe and completely healthy at all times.
 All plants, all animals, all organisms, all things, all species of life, are conscious people who have thoughts, feelings, desires, relationships, friendships, families, work, activities, hopes, dreams, wishes, willpower, bodies, spirits, minds, and souls. It is important to recognize everything and everyone as a conscious person who has a body, spirit, mind, and soul.
 It is also important to say: we heal very quickly. The immune system has many ways of removing harmful substances and the body has many ways of healing. We can cure a vitamin deficiency, or a mineral deficiency, very quickly by having a healthy and ethical diet. We can "detox" very quickly by having a healthy and ethical diet. Showering also helps us "detox", because it removes air pollutants from the body, and gives us more water to use in our biological processes. It is important that every person has safe food, safe drink, and safe air. A toxin is any substance that is poisonous. We have evolved to be very good at removing toxins. We have evolved to be very good at healing.