How these two Turing machine problems are different in terms of their decidability?
The thing you missed is that Turing Machines are strictly more expressive than Context-Free Grammars. CFGs are as expressive as push-down automata (PDA, i.e. automata with one stack).

E.g. the language $L = \{a^nb^nc^n | n &gt; 0\}$ cannot be expressed by CFG (or decided by a PDA), but can be decided by a Turing Machine.

So, for CFG, the emptiness problem is decidable, but somewhere on the way to Turing completeness, this property is lost.