The short answer is yes, they're different. Changes to power and toughness are effects that can be caused by a bunch of different things. Jarad gets a boost based on the number of creatures in the graveyard, there are abilities that let you spend mana for boosts (e.g. firebreathing), and there are spells that do it, typically until end of turn.
A +1/+1 counter is a counter that provides a +1/+1 effect. You can represent it by placing a physical counter of some kind on the creature. With Corpsejack Menace, you place twice as many counters.
Jarad's boost does not mention counters anywhere, so you do not get any benefit from Corpsejack Menace.