The first is a damage increase, adding X% of physical damage as fire damage. This works for *all* non-damage-over-time sources, including those from spells, attacks, and traps.
The second is the on-kill effect. Upon killing an enemy other nearby enemies are ignited for the value of "overkill" damage. The on-kill effect of Herald of Ash only applies to enemies killed by attacks.
"Overkill" damage refers to the damage of the killing hit which exceeds the remaining life of the monster.
(Example:If the player deals 1000 points of damage to an enemy with 400 points of remaining life the "overkill" damage would be 600 points)
The overkill damage value used for the on-kill effect is based on the total excess damage of the killing hit, not just its fire damage.