Bright blessings everyone. I am another pagan scientist witch. I was brought up in the far southwest England with an 'earth-based' spirituality that sees divinity in nature. I don't see anything contradictory in science and a pagan belief system. I wouldn't identify as Wiccan but will have a go at explaining the difference.
Paganism refers to a set of spiritual beliefs that revere nature and the changing seasons. It's often associated with polytheism (belief in many deities) or pantheism (belief in the spiritual or divine essence of all things). Wiccans are pagans in the way that Baptists are Christian, but not all pagans are Wiccan, just as not all Christians are Baptist.
Witchcraft is a skill or practice in the way that yoga is a skill or practice. It involves conducting material rituals and activities (possibly small, possibly very ceremonial) that you associate with an intention. The actions symbolise and are intended to manifest your desire. Some witches call on deities to help them manifest their will, but some don't. You don't have to be a pagan, wiccan, or to believe in any deities to practice witchcraft. Just as you don't have to be a specific religion to practice yoga or meditation, but some practitioners of religion will practice meditation as part of their spiritual practice. If you've ever made a wish whilst blowing out candles on a birthday cake, you've performed a small spell.
Wicca is a religion that is largely attributed to the work and actions Gerald Gardner (and others) in the 1950s. It is said to be based on old traditions (the 'old ways' of neolitchic Britain) but the evidence for this is disputed. Wicca is a form of paganism, but it is not the only form. There are many pagans that would not call themselves wiccan. Wiccans celebrate 8 holidays or Sabbats a year to mark the turning of the seasons, and view the divine as present in nature, and having both female and male elements. The moon is often considered a representation of the feminine divine and the 13 annual full moons (or Esbats) are times of power and sometimes celebrated alongside the 8 Sabbats.
Many Wiccans organise themselves in covens to celebrate the turning of the Wheel of the year together, to share ideas, research and their magical practice, but this isn't necessary I think. There are many solitary practitioners of Wicca.
So in short:
Pagans are not necessarily Wiccans or witches, but Wiccans would usually call themselves pagan. Paganism is an umbrella term.
Anyone can be a witch (I have known atheist, Christian, Wiccan, Pagan and Buddhist witches).
Wicca is a pagan religion. Wiccans often do practice magic/witchcraft but some do not.
I hope that helps! I'll speak more about my own practice if it is of interest..