  We report a systematic study of the superconducting (SC) and normal-state
anisotropy of Fe$_{1+y}$Te$_{0.6}$Se$_{0.4}$ single crystals with controlled
amounts of excess Fe ($y$ = 0, 0.07, and 0.14). The SC state anisotropy
$\gamma_{H}$ was obtained by measuring the upper critical fields under high
magnetic fields over 50 T for both $H\parallel ab$ and $H\parallel c$. On the
other hand, the normal state anisotropy $\gamma_{\rho}$ was obtained by
measuring the resistivity with current flowing in the $ab$ plane ($\rho_{ab}$)
and along the $c$ axis ($\rho_c$). To precisely measure $\rho_{ab}$ and
$\rho_c$ in the same part of a specimen avoiding the variation dependent on
pieces or parts, we adopt a new method using a micro-fabricated bridge with an
additional neck part along $c$ axis. The $\gamma_{H}$ decreases from a value
dependent on the amount of excess Fe at $T_{\rm{c}}$ to a common value $\sim$ 1
at 2 K. The different $\gamma_{H}$ at $T_{\rm{c}}$ ($\sim$1.5 for $y$ = 0, and
2.5 for $y$ = 0.14) suggests that the anisotropy of effective mass
$m_c^*/m_{ab}^*$ increases from $\sim$ 2.25 ($y$ = 0) to 6.25 ($y$ = 0.14) with
the excess Fe. The almost isotropic $\gamma_{H}$ at low temperatures is due to
the strong spin paramagnetic effect at $H\parallel ab$. By contrast, the
$\gamma_{\rho}$ shows a much larger value of $\sim$ 17 ($y$ = 0) to $\sim$ 50
($y$ = 0.14) at the temperature just above $T_{\rm{c}}$. Combined the results
of $\gamma_{H}$ and $\gamma_{\rho}$ near $T_{\rm{c}}$, we found out that the
discrepant anisotropies between the SC and normal states originates from a
large anisotropy of scattering time $\tau_{ab}$/$\tau_c$ $\sim$ 7.8. The
$\tau_{ab}$/$\tau_c$ is found to be independent of the excess Fe.
