With regards to Anthropology, what is the difference between the sciences and humanities and how are they interrelated?

Anthropology is a field of study which bridges the interesting gap between the sciences and the humanities. The study of humankind falls into the realm of "social science," or even "soft science," as opposed to what we might call the "hard sciences." The hard sciences are those like chemistry, geology, and biology, where knowledge is gained through rigorous observation and testing. Observation and testing take place in Anthropology as well, but here the results are far less predictable or standardized. The social science of Anthropology is markedly different from something like chemistry because humans introduce so many new variables to any situation. 

In the early days of Anthropology, many natural scientists were interested in learning about people from faraway locations or obtaining exotic artifacts. As Anthropology developed surrounding archaeological digs and intercultural relations, many early anthropologists wanted their studies to resemble the hard sciences. Unfortunately, much of the early study, measurement, and classification of peoples that took place was motivated by racist sentiment. Even today, some people attempt to use antiquated anthropological theories regarding race relations and "natural dispositions." As time and both the soft and hard sciences have progressed, the focus of anthropology passed through a phase of discovering what all people have in common. Though some anthropologists focus on this today, this question has been satisfactorily answered for many, so we turn our attentions towards trying to figure out when we humans came into existence. 

I don't want to be too reductive-- Anthropology covers a wide variety of interests! An anthropologist may study food, emotions, dance, language, gender, sexuality, home building, medicine and health, even how people throw out their trash! What all anthropologists have in common is that they try to incorporate the best of the hard and soft sciences into their studies. The hard sciences offer a quantitative perspective, meaning that an anthropologist may take concrete measurements for their studies. At the same time, the anthropologist considers the qualitative perspective-- why is their subject of study important? Sometimes, we phrase this in terms of the etic (measured by outsider) perspective and the emic (insider reasoning) perspective. Both are important parts of a holistic anthropological approach. 

As an example, a nutritional anthropologist who studies food practices might choose to learn more about how and why food is allocated in a family unit. In some parts of the world, the largest and best portions of food at meals go to the adult males, rather than dividing portions equally among family members. In the quantitative etic perspective, an anthropologist might note that an adult male eats 200g of meat per day, while children and women receive far less. Taking a concrete measurement like this offers insight into the kind of nutrition family members are receiving, but it doesn't tell us why the adult males get more meat. The qualitative emic perspective considers why it is important to families to give more meat to adult males. Perhaps the adult males perform physical labor and need more energy for their work. Perhaps there is a long-standing tradition of feeding fathers the best portions as a sign of respect. There are many possibilities and many variables! Even in two cultures which have similar practices of food allocation within families, their inside reasoning may be quite different. This is why it's so important for every anthropologist to incorporate the emic into their studies.