Damage? No. It may take longer for a given charge than the Note 4 charger does. Look at the output rating. It's around 5 Volts for both, but if the Amps (or mA) is too low for the S4, it'll take longer. The Note 4 should be able to draw 1.2A (or 1,200mA) during charge if it's available. If it's not (say the S3 charger can only supply 1A), the Note 4 will draw that amount until the battery is charged. (For instance, charging on USB is limited to about 900mA, but the phone charges nicely - it just takes longer.) It's been a LONG time since phone chargers put out anything other than 5.0-5.2 Volts, which is what matters. (And some of the old ones put out 4 Volts or 3.6 Volts, which wouldn't damage the phone, it just wouldn't charge the battery.)