In general Anthropology and Social/Cultural Anthropology refer to the same subject in the UK. In theory, Anthropology also includes Biological Anthropology, which considers human remains (e.g. study of human skeletons in either forensic scenarios or archaeological) and sometimes Archaeology (however that tends to be a subject unto itself here).
Sociology will consider more aspects of e.g. demography, social policy, social networks and more generally interactions on a group scale, as opposed to a societal/cultural scale as in Anthropology. Anthropology tends to consider more aspects of culture and social interaction on both larger and more intimate scales - ranging from cosmology/belief systems, concepts of kinship, the nature of culture as an abstract concept and so on. Things such as diasporan studies often have close ties to anthropology as the simple movement of those peoples between countries is not the only consideration but also how the culture of those people survives, perishes, or evolves.
Some Anthropology courses will include the elements of physical/biological anthropology and/or archaeology - at Oxford for example, it is only possible to study Anthropology through the Archaeology & Anthropology course, although there is no requirement to continue with either one beyond a certain point (I believe after second year, although I may be wrong). Conversely at Cambridge it's somewhat split, following the splitting of their degree of the same name - the Archaeology tripos includes archaeology and biological anthropology, and I think you can some external papers in social anthropology, and you can specialize in either direction (or in more specific areas like Egyptology I believe). The social anthropology part is contained in the HSPS (Human, Social, and Political Sciences) Tripos where as I recall at least two of the subjects have to be taken from social anthropology, sociology, and politics, with the possibility of an external paper from Archaeology, and subsequently more specialization is available. There is also Human Sciences, which is something of a rare course which tends to focus more on the biological aspects, both of anthropology and human biology generally, with an emphasis on human evolution usually. I also think at Exeter the Anthropology course also includes significant biological anthropology content. These are really exceptions to the rule however - the only other biological anthropology courses I can think of are at Oxford Brookes and Liverpool.
The best way to understand the differences is to look through the course content for each course - compare the modules/papers available in each course, which ones are core and which optional. You can then identify the course which has the most interesting content to you personally and naturally that would be the course set to apply to. It's possible you may be able to apply to courses from both sides with a single personal statement that has been very craftily written, but there is a risk it will not be relevant enough for either course and thus result in no offers being received.