Checking if a symbolic symmetrical Matrix is negative definite

Both of your approaches yield the same answer. it is the application of ToRadicals that causes the answer to be different. First, compare the two limits before the application of ToRadicals:

upperLimit1 = Sqrt[1 - Sqrt[a]];
upperLimit2 = Root[1 - a - 2 #1^2 + #1^4 &, 3];

Plot[upperLimit1, {a, 0, 1}]
Plot[upperLimit2, {a, 0, 1}]

upperLimit1 and upperLimit2 are the same over the region 0 < a < 1. Converting the Root object into radicals is problematic because the Root ordering depends on the parameter a. One suggestion would be to not use ToRadicals and just work with the Root objects. If you really want radicals, a naive application of ToRadicals:

ToRadicals[upperLimit2]

Sqrt[1 + Sqrt[a]]

is only correct for some values of the parameter a. However, in your case, you know something about the parameter a, so you should make use of that by giving ToRadicals an assumption:

ToRadicals[upperLimit2, Assumptions -> 0 < a < 1]

Sqrt[1 - Sqrt[a]]

Note that using:

ToRadicals[0 < a < 1 && upperLimit2]

0 < a < 1 && Sqrt[1 + Sqrt[a]]

does not cause ToRadicals to use 0 < a < 1 as an assumption. The assumption needs to be given explicitly as an option to ToRadicals.