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w2 = 6;
m = 2;
T = 2.0;
sol = NDSolveValue[{q'[t] == \[Zeta][t], \[Zeta]'[t] + w2*Sin[q[t]] ==
     0, q[0] == Pi/3., \[Zeta][0] == 0}, {q, \[Zeta]}, {t, 0, T}]

Plot[sol[[1]]'[t], {t, 0, T}

solFEM = NDSolveValue[{q'[t] == \[Zeta][t], \[Zeta]'[t] + w2*Sin[q[t]] == 0,DirichletCondition[{q[t] == Pi/3., \[Zeta][t] == 0}, 
 t == 0]}, {q, \[Zeta]}, t \[Element] Line[{{0}, {T}}], Method -> {"FiniteElement"}]

Plot[solFEM[[1]]'[t], {t, 0, T}

 solFEM = 
 NDSolveValue[{q'[t] == \[Zeta][t], \[Zeta]'[t] + w2*Sin[q[t]] == 0, 
   DirichletCondition[{q[t] == Pi/3., \[Zeta][t] == 0}, 
    t == 0]}, {q, \[Zeta]}, t \[Element] Line[{{0}, {T}}], 
  Method -> {"FiniteElement", 
    "MeshOptions" -> {MaxCellMeasure -> 0.01}}]

In[26]:=FindMinimum[sol[[1]]'[t], {t, 0.5}]
Out[26]={-2.44949, {t -> 0.688223}}

In[27]:=FindMinimum[solFEM[[1]]'[t], {t, 0.5}]
 
••• FindMinimum: The line search decreased the step size to within the tolerance specified by AccuracyGoal and PrecisionGoal but was unable to find a sufficient decrease in the function. You may need more than MachinePrecision digits of working precision to meet these tolerances.

Out[27]={-2.1981, {t -> 0.5}}

In[28]:=FindMinimum[solFEM[[1]]'[t], {t, 0.688223}]
 
••• FindMinimum: Line search unable to find a sufficient decrease in the function value with MachinePrecision digit precision.

Out[28]={-2.45211, {t -> 0.69}}

w2 = 6;
m = 2;
T = 2.0;
sol = NDSolveValue[{q'[t] == ζ[t], ζ'[t] + w2*Sin[q[t]] == 0, 
                    q[0] == Pi/3., ζ[0] == 0}, {q, ζ}, {t, 0, T}]

Plot[sol[[1]]'[t], {t, 0, T}]

solFEM = NDSolveValue[{q'[t] == ζ[t], ζ'[t] + w2*Sin[q[t]] == 0,
                       DirichletCondition[{q[t] == Pi/3., ζ[t] == 0}, t == 0]}, 
                      {q, ζ}, t ∈ Line[{{0}, {T}}], Method -> {"FiniteElement"}]

Plot[solFEM[[1]]'[t], {t, 0, T}]

 solFEM = 
 NDSolveValue[{q'[t] == ζ[t], ζ'[t] + w2*Sin[q[t]] == 0, 
   DirichletCondition[{q[t] == Pi/3., ζ[t] == 0}, t == 0]}, 
   {q, ζ}, t ∈ Line[{{0}, {T}}], 
   Method -> {"FiniteElement", "MeshOptions" -> {MaxCellMeasure -> 0.01}}]

FindMinimum[sol[[1]]'[t], {t, 0.5}]
(* {-2.44949, {t -> 0.688223}} *)

FindMinimum[solFEM[[1]]'[t], {t, 0.5}]

••• FindMinimum: The line search decreased the step size to within the tolerance specified by AccuracyGoal and PrecisionGoal but was unable to find a sufficient decrease in the function. You may need more than MachinePrecision digits of working precision to meet these tolerances.

(* {-2.1981, {t -> 0.5}} *)

FindMinimum[solFEM[[1]]'[t], {t, 0.688223}]

••• FindMinimum: Line search unable to find a sufficient decrease in the function value with MachinePrecision digit precision.

(* {-2.45211, {t -> 0.69}} *)

EDIT 1

On appearance, an easy fix would have been to reduce the cell size of the FEM method. This can be achieved by setting the "MaxCellMeasure" size to something smaller (like 0.01 or 0.001 for example):

 solFEM = 
 NDSolveValue[{q'[t] == \[Zeta][t], \[Zeta]'[t] + w2*Sin[q[t]] == 0, 
   DirichletCondition[{q[t] == Pi/3., \[Zeta][t] == 0}, 
    t == 0]}, {q, \[Zeta]}, t \[Element] Line[{{0}, {T}}], 
  Method -> {"FiniteElement", 
    "MeshOptions" -> {MaxCellMeasure -> 0.01}}]

However, this only "hides" the issue to the bare eye. One can still see the chattering effect when zooming around the minimum for example:

Why would this matter?

My problem with this issue has nothing to do with the curve appearance. On a practical level, if I want to determine, say, the local minimum (or maximum) near some region, then FindMinimum (or FindMaximum) will inevitably fail. I can easily find the local minimum near t->0.5 for the "default" solution:

In[26]:=FindMinimum[sol[[1]]'[t], {t, 0.5}]
Out[26]={-2.44949, {t -> 0.688223}}

However, this is not possible with the FEM solution because chattering obviously affects FindMinimum:

In[27]:=FindMinimum[solFEM[[1]]'[t], {t, 0.5}]

••• FindMinimum: The line search decreased the step size to within the tolerance specified by AccuracyGoal and PrecisionGoal but was unable to find a sufficient decrease in the function. You may need more than MachinePrecision digits of working precision to meet these tolerances.

Out[27]={-2.1981, {t -> 0.5}}

Because of chattering, this simple operation becomes impossible even when directly feeding a very close initial guess:

In[28]:=FindMinimum[solFEM[[1]]'[t], {t, 0.688223}]

••• FindMinimum: Line search unable to find a sufficient decrease in the function value with MachinePrecision digit precision.

Out[28]={-2.45211, {t -> 0.69}}

This is only one example of the consequences of that chattering effect.