Mindfulness meditation is a term we use to describe the practices that certain famous/influential western meditation practitioners developed after coming back from the mostly theravada countries that they went to to learn meditation practices from the monks there in the 50-70s. Generally speaking, those practices were vipassana/insight practices, and so mindfulness practices are in essence, or at least are modified vipassana techniques. Mindfulness is the translation we tend to use for the Pali `sati`, mindfulness techniques tend to develop this quality of `sati`, which is a kind of re-contextualisation of your perception from "I am having an experience" to "an experience is happening within conscious awareness" in a direct, first person way. And yeah, the development of sati, along with other helpful factors, can help to gain insight, which is basically the way the mind looks at itself and conceptualises its own experience, in a kind of self-reflective way, and what it means to relax and tighten the tension that holding the mind in that way causes.

In these western teachings and in the theravadan writings, usually insight is contrasted with "concentration", which is `samatha`, which is a kind of effortless concentration on one thing. Focusing solely on the sensations of the breath at a point in your body is one way of doing samatha, or imagining a complex mental image (of, say, a mandala) or a simple image like a white circle. This kind of meditation is about dropping your attachment to everything except this one thing, which creates a strong attachment to that thing to the exclusion of all else, and this is often pleasurable because you're both strongly mentally attached to that sensation and it's a sensation that keeps happening at high "volume", which is very rewarding. With this you can enter states called `jhana` (which is actually pretty much the only word in early buddhism for "meditation", and through the spread of Buddhism, linguistically became "chan" in China, "zen" in Japan, and "seon" in Korea, "thien" in Vietnam...)

The reason that they are contrasted is because mindfulness practices tend to go the other way and include a larger scope of what you are including in your attention, for example mentally labelling what senses are being activated from moment to moment, or the sensations of the body/breath in a more wholesome way without blocking out external stimuli, but developing very strong clarity on the momentary aspects of those sensations and experiences, sort of applying a "samatha" style concentration to analysing the sensations as the "three core features" you described.

Both the approaches tend to suppress the "hindrances" in the mind that prevent jhana and insight. These are recurring thoughts and emotions that trouble you and stop this kind of mental configuration from happening in a stable way. The techniques use up mental and perceptual "bandwidth" in holding the mind in a way so that attention/awareness and holding on/letting go are balanced on whatever the object of meditation is. By developing samatha or sati, or a some of both mixed together, you can enter interesting and unusual states of consciousness that are pleasant, weird, intriguing, peaceful- that can be used as a point of inquiry into why the mind is grasping at things in certain ways and what the result of that grasping is doing to it.