difference between Simplify and FullSimplify

I try to make the following formula as simple as possible.

(\[Theta]*Subscript[\[Alpha], 1]*(2 + (1/2)*(\[Theta]*Subscript[\[Alpha], 1] +        Sqrt[8*\[Theta]*Subscript[\[Alpha], 1] + \[Theta]^2*Subscript[\[Alpha], 1]^2])))/   ((1/2)*(\[Theta]*Subscript[\[Alpha], 1] + Sqrt[8*\[Theta]*Subscript[\[Alpha], 1] +         \[Theta]^2*Subscript[\[Alpha], 1]^2]) + \[Theta]*Subscript[\[Alpha], 1]*     (2 + (1/2)*(\[Theta]*Subscript[\[Alpha], 1] + Sqrt[8*\[Theta]*Subscript[\[Alpha], 1] +           \[Theta]^2*Subscript[\[Alpha], 1]^2]))) 

So. I use Simplify and get the result.

(\[Theta]*Subscript[\[Alpha], 1]*(4 + \[Theta]*Subscript[\[Alpha], 1] +      Sqrt[\[Theta]*Subscript[\[Alpha], 1]*(8 + \[Theta]*Subscript[\[Alpha], 1])]))/   (\[Theta]^2*Subscript[\[Alpha], 1]^2 + Sqrt[\[Theta]*Subscript[\[Alpha], 1]*      (8 + \[Theta]*Subscript[\[Alpha], 1])] + \[Theta]*Subscript[\[Alpha], 1]*     (5 + Sqrt[\[Theta]*Subscript[\[Alpha], 1]*(8 + \[Theta]*Subscript[\[Alpha], 1])])) 

This is the thing I can calculate by hands as well. However, if I useFullSimplifythen it gives

(4*\[Theta]*Subscript[\[Alpha], 1])/(3*\[Theta]*Subscript[\[Alpha], 1] +     Sqrt[\[Theta]*Subscript[\[Alpha], 1]*(8 + \[Theta]*Subscript[\[Alpha], 1])]) 

which is more simple. However, I have a fundamental suspicion about the result from FullSimplify. Thus, I check these two are indentical.

sol3 = (\[Theta]*     Subscript[\[Alpha], 1]*(4 + \[Theta]*Subscript[\[Alpha], 1] +         Sqrt[\[Theta]*         Subscript[\[Alpha],           1]*(8 + \[Theta]*Subscript[\[Alpha], 1])]))/      (\[Theta]^2*Subscript[\[Alpha], 1]^2 +      Sqrt[\[Theta]*Subscript[\[Alpha], 1]*             (8 + \[Theta]*Subscript[\[Alpha], 1])] + \[Theta]*      Subscript[\[Alpha], 1]*           (5 +         Sqrt[\[Theta]*          Subscript[\[Alpha],            1]*(8 + \[Theta]*Subscript[\[Alpha], 1])])) 

and

sol4 = (4*\[Theta]*Subscript[\[Alpha], 1])/(3*\[Theta]*Subscript[\[Alpha], 1] +      Sqrt[\[Theta]*Subscript[\[Alpha], 1]*(8 + \[Theta]*Subscript[\[Alpha], 1])])    {sol3, sol4} // RootReduce     SameQ @@ (Sort /@ %)  

which gives False. I don’t know how could we get FullSimplify result from the Simplify result.