When I look at the Wikipedia entry for biostatistics, the relation to biometrics doesn't seem so obvious to me since, historically, biometrics was more concerned with characterizing individuals by some phenotypes of interest, with large applications in population genetics (as exemplified by the work of Fisher), whereas part of this discipline now focus on biometric systems (whose objectives are the "recognition or identification of individuals based on some physical or behavioral characteristics that are intrinsically unique for each individual", according to Boulgouris et al., Biometrics, 2010). Anyway, there still are reviews like Biometrika and Biometrics; although I read the latter on an irregular basis, most articles focus on "biostatistical" theoretical or applied work. The same applies for Biostatistics. By "biostatistical" applications, I mean that it has to do with applications or models related to the biomedical domain, in a wide sense (biology, health science, genetics, etc.).
According to the Encyclopedia of Biostatistics (2005, 2nd ed.),
(...) As is clear from the above examples,
biostatistics is problem oriented. It
is specifically directed to questions
that arise in biomedical science. The
methods of biostatistics are the
methods of statistics -- concepts
directed at variation in observations
and methods for extracting information
from observations in the face of
variation from various sources, but
notably from variation in the
responses of living organisms and
particularly human beings under study.
Biostatistical activity spans a broad
range of scientific inquiry, from the
basic structure and functions of human
beings, through the interactions of
human beings with their environment,
including problems of environmental
toxicities and sanitation, health
enhancement and education, disease
prevention and therapy, the
organization of health care systems
and health care financing.
In sum, I think that Biostatistics is part of a super-family--Statistics--, and share most of its methods, but has a more focused area of interest (hence, an historical background, specific designs, and a general theoretical framework) and dedicated modeling strategies.