丢脸: literally, lose your face
丢人：literally, lose your whole life
丢人 is slightly more serious than 丢脸
when you blame someone(offending):
when you blame yourself:
我太丢脸了！less widely used
when you blame a specific thing or action:
这件事太丢脸了！ok but less widely used
so all the cases above, you'd better use 丢人 instead of 丢脸
to feel ashamed:
to make someone feel ashamed:
you may do something wrong to make your father feel ashamed(丢脸) of you, but event if you had ruined the whole world, you cannot make your father feel 丢人
丟人 is only used on the person doing things that is humiliating. I would translate it as "humiliating","degrading"
丟臉 can be used on the one related to this humiliating things. I would translate it as "embarrassing/embarrassed", "disgraceful/shameful"
I agree with others. 丢人 could be a bit more serious than 丢脸, but since the difference is not obvious, and the "seriousness" can be quite subjective, people may use either one in their statements.
For example, you can find many people debating this issue.