I see the answers here just define the domain of work so I try to give a more comprehensive answer based on my experience of learning statistics as a medical practitioner. Most of my experience is on clinical trials, but this can be applied to any domain of biostatistics.
The purpose of biostatistics is biological and medical field, this gives it subtle differences according to this purpose.
Statistics is all the same! it is just math! However, here is the difference that comes to my head when I define biostatistics.
1- Ordinary statistician will not understand all the terminologies in biostatistics but he will understand the math!
Both of them are coming from mathematical and probability theories. So you will find most of the tests resonates will with both words like regression analysis, t-test ... etc
However, when it comes some other tests like relative risk, attributable risk reduction, kaplen mieir curves ... etc these few tests will sound strange for someone with no biostatistical knowledge. However, they can easily go through it when they read about these tests
2- Biostatistics field usually don't reinvent the wheel, they just enhance what is available
As I said biostatistics is built on statistics. But unlike the previous point, most of the current active research on biostatistics is mostly about enhancing few properties of existing test with different terminology to serve the purpose of biostatistics. For example, something like overall survival or time-to-death are all terminologies exclusive for biostatistics (that's for sure or who would study life and death) however they are built on time-to-event analysis that biostatistician has created these terminologies to make the test serve the purpose of biostatistics, more standardized and easy to interpret in among medical practitioners.
3- Biostatistics has its specific guidelines (just like any other field) however it is more strict.
Biostatistics has established many guidelines and conventions to analyze the data of different field. For example, statisticians working in biology and genomics are doing different tests and have different thinking than who are working in clinical trials(and of course who are working in business intelligence). But this way of working is considered fixed among the community of biostatistician, so a biostatistician don't usually think out of the box unless there is something urges that has not existed before, and this usually don't happen as study design of biostatistics fields is very definitive.
A clearer example of this is the baysian statistics application on biostatistics. Bayesian statistics are known to be flexible, so you will not find a lot of usage of this type of statistics. Also, this usage is tied to a certain repetitive application like sensitivity measurement. There is no need to think of probabilities when there are easier options that are easier to interpret and perform.
Why This restriction?
1. The community is trying to avoid p hacking and beautifying the results. Especially if you are working in clinical trials, you don't just use the tests the gives the best results. You even don't use one-sided tests usually! These conventions are there to protect the trials validity and anything else will make the community suspicious.
That's the most important part. All the work of biostatistics should be interpreted by a medical practitioner, so he should make some sense of results himself. So they try to stick to a few approaches.
This point is unfair because there is no comparison, but study design in biostatistics is very definitive. Usually, you don't have to think a lot on how to prove the efficacy of a drug or adverse effect or so. So it is very unlikely you will need to keep your head busy of learning different techniques and tests every while as it is very rare to see a pattern change.
That's all I have right now, I will update my answer if I remembered something else.