Even though you have not seen the affair you assume that there is an illicit affair between your mother and the neighbour. probably you are right, assuming your hunch is right i have to say the following.
All are humanbeings and o one is perfect in this world. as human beings every one has desires. sex is a basic need of human body, also of mind. if the need is not met, life is difficult for all.( you have not mentioned your age, still i hope you can understand that our bodies are biological machines, with it's needs to be met)
A woman in her 40s is highly in need of sex, as i understand, whether it is my wife, sister or mother. and sex is not dirty or it not sin to engage in sex.
probably your mother is notggetting enough sexual satisfaction, her body/biology demands.
So please try to see this situation in a human and also humane angle. she has not acted, against the interest of her family, yet.
it is natural that you are upset, but see that she is also a human being. there are many husbands and wifes in this world who are forced to seek sexual satisfaction outside the marriage, due to circumstances.
At any rate do not tell any one about the matter. if possible, looking at the situation with sympathy she deserves, try to forget the matter. if you look at in a biological perspective, it is not a big thing, i know these things happen in the world, as we all are human beings.
if you want you talk to your mom about it, but not in a way of accusing her of committing a great sin. please do not make her feel guilty, which will make her life miserable. as a mature woman she she will know what to do in future, once you talk to her.
at any rate do not tell your father. mistakes can be corrected, if you show maturity, understanding, and kindness.
once you grow up you will know that these things are pardonable.
let me remind you that no one is perfect and no situation in life is perfect. there are no right or wrong. it all depends on how we look at things. but we tend to attach too much importance to one's sexual activities.
there are many things we do not know about her family/marital life. i answered your question assuming you are a very young man or teenager, unmarried person. had you been a mature person( age wise) you would have kept silence, after assuring that the family life/ peace is not affected.
hope your peace of mind is restored soon.