We don't know if there is a soul or not. There is no evidence that one exists. From what we do know, the brain SEEMS to be the seat of consciousness. Every single experiment and observation indicates that this is the case.
Your memories are stored in your brain. How do we know this? Well we have examined patients that have had brain damage to certain regions of the brain, and have lost memories. Logically, this indicates that the brain houses your memories as a pattern of neural connections.
Your ability to process images, understand language, count, imagine 3D objects, and even recognize faces are all known to be different functions of the brain. Again, medical science has case histories of people who have lost these abilities after having their brains damaged by accident or disease.
There have been accounts of individuals undergoing rapid personality shifts when their brains are damaged (The case of Mr. Gage is an example). Again, this indicates the likelihood that your personality, memory, cognitive functions and whatnot are functions of the neural tissue of your brain.
Does this knowledge absolutely rule out the possibility of a soul..... well no. But we are learning more and more about the brain every single day, and it is becoming ever more likely that consciousness is just another function of the brain, albeit a function that involves multiple sections of the brain working in tandem.
So although science cannot (yet) rule out the possibility of the soul, there is no evidence of one. In my opinion, the "soul" was a concept created by people a long time ago, who didn't know much about neural science and biology. The concept of a soul was just an ad-hoc explanation for the phenomena of consciousness, which the ancients, in their ignorance, could not explain.
I fail to see why people should cling to old ideas, when the new ideas fit the evidence better. Simply put, you have no reason to believe in the soul, other than the fact that you want to believe in it. And merely wanting something to be true is never a good reason for believing it to be true.
Yes, science cannot explain everything. It is a human processes that requires effort. But if you are going to use the fact that science cannot explain everything to justify religion, that's a logical fallacy. It's a "god-of-the-gaps" - where you try to cram in your god for every single question we don't currently have a definitive answer for. As science closes these gaps, as it has been doing for hundreds of years, the areas for people to let their god dwell shrink rapidly, ever so.
500 years ago, we didn't know how the rain formed, so we said "a god did it". Now we know it is due to water vapour condensing in the atmosphere. A long time ago, we didn't know what the sun was, so many people said it was a god. But now we do know what the sun is - it is an immense ball of (mostly) hydrogen gas undergoing nuclear fusion.
Science doesn't know everything, and sometimes it gets things wrong. But we learn more and more and our theories get ever more accurate. Proof that science is right in most things is all around you in the form of technology. If our theories on electromagnetic forces and atomic physics was incorrect, you wouldn't be using a computer, and a nuclear power plant would fail to produce energy.
Science has, through technology, proven itself to be a very good way of understanding the universe. Because it relies on human knowledge and human experimentation, mistakes will be made. But science allows for the correction of mistakes.
Religion, on the other hand, admits no errors or mistakes. It insists that it is perfect from the get-go. It insists that it is divinely inspired. So, unlike science which is a human effort, the Bible should have NO MISTAKES in it because it supposedly comes from a god/gods. Yet it does have mistakes - the genesis account of creation is ludicrous and outdated. Noah's ark is a biological and physical impossibility. Humans cannot walk on water. You can claim that this is "god's miracles", but that suspiciously sounds like "magic" to me.
Simply put, science explains much more than religion, and provides much more power than religion. You say that science cannot explain everything, but "god did it" is HARDLY an explanation, but merely a way to ignore the problem. Science tries to find answers that can be backed by evidence. Religion just says "god did it, the end", as if that was a courageous way of facing the world.
Religiosity is the opposite of curiosity. Science explores and creates, while religion merely claims it is true through sheer force of insistence and faith.
I don't know if there is a god. But logically, if I have no proof, to believe in one would be silly. I can only believe in things that can be proven, and science has a lot of proof backing itself up. Religion can only hid in the gaps in our knowledge, which are ever shrinking. One day, maybe, there will be no gaps. Where then will religion hide?