Difference between multinomia logistic regression and N 1 vs all binary
Quote from Alan, Agresti. "Categorical data analysis." A John Wiley and Sons, Inc. Publication, Hoboken, New Jersey, USA (2002), page 273:


  The separate-fitting estimates differ from the ML estimates for simultaneous fitting of the J-1 logits. They are less efficient, tending to have larger standard errors. However, Begg and Gray 1984 showed that the efficiency loss is minor when the response category having highest prevalence is the baseline.