It doesn't matter what comes before the word yoga: All yoga traditions aim to calm the mind (2nd Yoga Sutra)!

The modern system of  Hatha yoga  includes a large amount of health-promoting physical exercise. Everyone is physical yoga  Hatha yoga.
There are exercises (asanas) for the back and spine, strengthening the muscles, exercises to promote a healthy metabolism, exercises to calm the autonomic nervous system, etc., etc. - this list could go on and on for a long time.

The most important expression of energy in the body (prana) is breath! 
Thus, im  Hatha yoga  conscious breathing is the center of every practice. The physical breathing movement is comparable to the flywheel of a machine. If you "master" this drive wheel, you can begin to penetrate into ever finer areas of your being until all levels gradually become conscious and finally the great secret of life is revealed.

The origin of  Vinyasa yoga  is  Ashtanga (Vinyasa) yoga.
Ashtanga Vinyasa Yoga  is an old yoga tradition in which 4 different series (sequence of yoga positions) are taught. These are the same all over the world!
The most common is the first series or parts of it. The goal of the first series is to restore or maintain general physical health. The sequence of yoga positions has been "designed" to cleanse and strengthen the body.
The essential and characteristic basic building block of this dynamic yoga style is the breathing technique (ujjayi), which is the central element of this yoga practice. The sequence of movements is linked to breathing. In addition, the activation of certain energy centers (bandha), the yoga positions (asana), the optical focus (drishti), the sequence of movements (vinyasa) are important components of this practice.

The system could be thousands of years old. It is not known how long it has been handed down orally. The oldest written source for this  Ashtanga yoga system  is probably the manuscript "Yoga Korunta" (Sanskrit for "yoga groups") by Vamana Rishi, who is said to have written it on palm leaves centuries ago.

Vinyasa yoga  is that  Asthtanga yoga  accordingly an expression of absolute precision and a lively, free-spirited flow! Old yoga tradition is combined with new innovative thinking and enables "today's people" a new attitude towards life. 
Vinyasa yoga  is the modern and creative variant of the  Ashtanga Vinyasa Yoga. There are no fixed series here.
Based on the same technique, each yoga class creates a new, thematically structured sequence of yoga positions that arise from the flow of breath and flow into one another. 
The yoga teacher tailors the selection of topics to the needs and level of knowledge of his students.
This wealth of variants makes  Vinyasa yoga  to a very modern, effective and satisfying yoga practice.