It is a consequence of
In[10]:= Norm[{a, b}]
Out[10]= Sqrt[Abs[a]^2 + Abs[b]^2]
where a is Sin[t] + t*Cos[t] and b is 2 t.
You may get a result closer to the expected one by specifying assumptions, for example
In[11]:= Simplify[Norm[{(Sin[t] + t*Cos[t]), 2 t}], Element[t, Reals]]
Out[11]= Sqrt[4 t^2 + (t Cos[t] + Sin[t])^2]
In[12]:= Expand /@ %
Out[12]= Sqrt[4 t^2 + t^2 Cos[t]^2 + 2 t Cos[t] Sin[t] + Sin[t]^2]
