These bonds both contain embedded options but the investors position is different. In the putable bond, the investor essentially has a position equal to a long position in a plain vanilla bond plus a LONG position in the put. So, the increased value of the put from an increase in volatility increases the value of the investor’s bond+Put portfolio.
In contrast, In the callable bond, the investor’s position is equivalent to a long position in a plain vanilla bond plus a SHORT position in the call. If the call is exercised, the bond issuer benefits (and since it’s a zero-sum game, the investor loses). So, while the value of the call increases with higher volatility, it DECREASES the value of the investor’s (bond-call) portfolio.