Statistics vs. Biostatistics does not make sense as a comparison; biostatistics is really a sub topic of statistics. This would be like asking "what's the difference between mathematics and probability?"; probability is a subfield of mathematics.
As others have noted, biostatistics appliedapplies to problems that are very common in both medical studies and biological research. This includes, but certainly is not limited to, survival analysis, sequential trial design, longitudinal analysis and genomic analyses, to name only a few topics.
As for the difference between programs in statistics and biostatistics, the obvious difference between two programs is that the biostatistics programs will be specializing in the topics above. Most statistics programs will still cover biostatistics (for example, I have my PhD in Statistics, and of all possible specializations of statistician, I am most qualified as a biostatistician, my current position), but it is definitely possible to get a PhD in statistics with only a mild introduction to biostatistic-specific topics.
It's my understanding that the high demand for statisticians by pharmaceutical companies lead to the demand for biostatistics programs.