The difference and similarity between Poisson and BM

The more I learn about BM, the more I feel there are strong connection between Poisson and BM. Last week, I ask a question. If we discrete Poisson process to small time interval, it behaves exactly as random walk with probablity lamda*h up by one and 0 otherwise. And counterpart of BM in discrete world is also random work with probability 0.5 up by 1 and probability 0.5 down by 1. Then I ask a question, what if we change the measure just like that we can change the BM with drift to BM without drift. Answer is obvious no, there are several differences.

• discrete BM, up and down probability is fixed. the only scale factor is step size. But the scale factor of poisson process is probability up and step size is fixed
• when we change the measure, we require that two probability measuresare equivalent which means the null space is the same. The null space of discrete one step of randome walke is R/{-1,1}. But poisson is R/{0,1}.

But when I ask Professor what is discrete case of BM with drift. Is it with the symmetric step size but different probability or different step size but the same probability or both or does not matter. I have not get answer yet.