Blank tape halting problem vs. Emptiness problem ($H_0$ vs. $E_{TM}$)
Yes, they are different. $L(M)$ is the set of all inputs accepted by a Turing machine, so $E_{TM}$ asks if a Turing machine doesn't accept any inputs. $H_0$ asks whether a particular input (the empty string) is accepted. BTW, $E_{TM}$ should be co-Turing-recognizable, but not Turing-recognizable.