It is not at all unusual for children to have different impressions of a parent’s situations and needs and of the best care for her. Often this has to do, as you suggest, with the amount of time they spend with the parent. It can often help to involve a professional geriatric care manager who can provide an outside and educated perspective, as well as advise on local care options. Also, everyone needs to be aware that dementia is not static and can get worse over time.
That said, as long as your mother has not been declared incapacitated by a court she has the same rights you or I have to decide where she lives. While she can request your assistance and delegate certain matters to you, without a court appointment you can’t make a residential decision for your mother. This is something of a gap between durable powers of attorney – which allow the delegation of legal and financial authority – and health care proxies – which permit the delegation of health care decisions. So, for instance, under a durable power of attorney your brother can sign a contract so that your mother has a room at an assisted living facility, but neither he nor you can force her to move there.