Solve[ ] with Method -> Reduce gives a different result than Reduce[ ]

Question

Why does

Solve[Sqrt[x + Sqrt[x]] - Sqrt[x - Sqrt[x]] ==
      m Sqrt[x/(x + Sqrt[x])], x, Reals, Method -> Reduce]

give a different result than

Reduce[Sqrt[x + Sqrt[x]] - Sqrt[x - Sqrt[x]] ==
       m Sqrt[x/(x + Sqrt[x])], x, Reals]

note that the first one says $1<m<2$ while the second one says $1<m\le 2$.
The documentation says

With Method->Reduce, Solve uses only equivalent transformations and finds all solutions.

Answer 1

Solve by default works with generic parameters, even if you use the option Method -> Reduce. To get the special parameter value m = 2 you need to set MaxExtraConditions to All:

Solve[Sqrt[x + Sqrt[x]] - Sqrt[x - Sqrt[x]] == m Sqrt[x/(x + Sqrt[x])], x, Reals, 
    Method -> Reduce, MaxExtraConditions -> All]
Out[1]= {{x -> ConditionalExpression[(4 - 8 m + 8 m^2 - 4 m^3 + m^4)/(4 - 8 m + 4 m^2),
    1 < m <= 2]}}