These two words are so misused that it is difficult to distinguish their meanings. Many people appear to use both terms interchangeably, while 'inhuman' is frequently used where I would use 'inhumane', and vice versa.
Because of the degree of confusion between these two terms, I will not claim my own usage as being the 'correct' one, however, I will attempt to explain how I differentiate them.
To me, it is less severe to call someone 'inhumane' than to call them 'inhuman'. I would use 'inhumane' to describe someone who is indifferent to the feelings, sensibilities, or dignity of other people (or sometimes a specific group of people). In some cases, this may also extend to insensitivity towards animals. e.g. It is inhumane to lock refugees away in camps with inadequate sanitary conditions and poor access to schools, medicine, and contact with the outside world. It is 'inhumane' to raise animals in overcrowded cages, where they can hardly move, just so we can reduce the cost of food for humans.
I would use 'inhuman' to describe someone who inflicts senseless cruelty on another, especially if the perpetrator gets some form of sadistic enjoyment from such actions. e.g. It is 'inhuman' to torture, starve, rape, or mutilate other people, or to indiscriminately destroy their homes, places of work, or their communities.
To me, an 'inhumane' person and an 'inhuman' person both lack the emotions of kindness, pity, or compassion towards other people. However, 'inhuman' people take this a step further, to where they can perform acts of barbarity upon others that exceed the expected bounds of human behaviour.