The simplest answer is that 'neither' means 'not either' (and 'nor' means 'not or'), and should be used accordingly:
'I cannot get to Tokyo on either the 5:15 or the 6:15 train.'
'I can get to Tokyo on neither the 5:15 nor the 6:15 train.'
They sometimes function slightly differently:
I can't get to Tokyo today, and ne