Well, the way you said it isn't quite right. You could spend the same $Delta V$ to escape Earth in one direction as the other, and in one direction you would move away from the Sun, and in the other direction you would move towards it.
What I think you're referring to is the fact that it would take far less $Delta V$ from Earth to escape the Solar System than it would to reach the Sun and touch its surface. From low-Earth orbit, it takes $8.75,mathrm{km/s}$ to escape the Solar System completely, but it takes $16.8,mathrm{km/s}$ to dive into the Sun from low-Earth orbit.
Of course, you wouldn't do that $Delta V$ yourself. In either direction, you would use Jupiter to send you in or out (or perhaps even in for the out case, if you want to do an escape maneuver near the Sun, which can be very efficient).
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