The two results are very close to each other. If you calculate

a = Integrate[
   Exp[(Cos[\[Pi] x] - 1)/(2 \[Pi])] Cos[6*\[Pi]*x], {x, 0, 1}];
N[a]
N[a, 16]

you will see somewhat different results. This is probably due to the different methods used by N[a] and N[a,16]. I understand that N[a] uses machine precision from the start, and the precision can degrade quite a bit. N[a,16] imposes that the end result has 16 accurate digits, and chooses the starting precision for that goal. It seems that NIntegrate in this case is more precise than N[Integrate] at machine precision.