Stillness lies beyond mindfulness
Stillness
Ainslie Meares began using the term Stillness in relation to meditation more than 50 years ago eg Stillness, stilling the mind etc. He is the source of several terms in common use today including Stillness. Some people who use the term "stillness" have no relationship to Meares' method. They often use the term differently as in non-stillness meditation the mind retains activity rather than being completely still.
Mindfulness
The word "mindefull" has been in use for several hundred years. Last century it was applied to certain spiritual beliefs to translate them into English. The body of research on mindfulness has not been on one type of meditation but on many types. Some meditation schools even changed their names to mindfulness so they could be associated with the research. Those with the spiritual belief mentioned earlier, practice tolerance of other's views, but likely still feel unsettled by this usage. These days, mindfulness meditation refers to focussing on one thing (eg present moment), a mental monotone.
Stillness lies beyond mindfulness
A full mind is not empty or still. Being mindful of one thing is not an empty or still mind. The focus or narrowing to a monotone prevents the mind from becoming still. Of course, if the focus pracittioner accidentally "forgets" their focus then they may slip into Stillness. This is quite distinct from learning a direct approach that allows the mind to slow and still.
In Stillness Meditation there is no focus on breathing. No visualization. No mantras. No chanting. No candle gazing. No thinking or focus on feeling of compassion or other emotion (during meditation practice). Meares' method is unrelated to other types of meditation. It involves is a really simple and natural experience. There is no thought, no sensation and no emotion when the mind is still. You do know that you are not asleep nor unconscious but remain dimly awake. It is afterwards that you know of the calm and mind-body rest. You will know what is meant after you have tasted it. Sometimes, the calm is deeper than calm, like tranquility or even serenity.