Perhaps the problem is that the expression

E^(\[Alpha] ((2 Log[-\[Delta] \[Lambda]T])/(\[Rho] - \[Alpha])))

is replaced with

(-\[Delta] \[Lambda]T)^((2 \[Alpha])/(-\[Alpha] + \[Rho]))

before Simplify has a chance to work, and this expression is a little too complicated. A workaround is ti Inactivate the Log, so that the replacement does not take place:

Inactivate[
  E^(-\[Alpha] (t + 4 T - (
      2 Log[-\[Delta] \[Lambda]T])/(\[Rho] - \[Alpha]))) -
   E^(-\[Alpha] (t + 4 T)) *
    E^(\[Alpha] ((2 Log[-\[Delta] \[Lambda]T])/(\[Rho] - \[Alpha]))),
  Log] // Simplify