Everyone seems to be asking this, so I'll just list a few relevant differences. I haven't even played with either of these abilities, so if I'm missing anything/wrong about Replicate someone should tell me.
Let's begin with an example, using the latest rumored spoiler (name is made up)
Giant Giant Growth Growth 1G
Sorcery
Target creature gets +3/+3 until end of turn
Multikick 1G: Target creature gets +3/+3 until end of turn
With Replicate instead of Multikick, these spells would do very similar things. You can target either the same creature or multiple creatures with each instance. You can pay 1G as many times as you want to get the repeated effect: 3GGG for a total of +9/+9 P/T boost, 4GGGG for +12/+12, etc.
However:
Replicate copies the spell, and therefore multiple spells go on the stack during resolution. That means each copy needs its own counterspell in order to be fully countered. Or you can Mindbreak Trap to exile all the instances on the stack at once.
With Multikick, you only need one counterspell to get rid of the whole effect.
Replicate only appeared on Instants and Sorceries.
Multikick can appear on permanents, which opens up a lot of design space. (So far, we know that multikick will be used to make creatures bigger upon entering the battlefield.)
Replicate creates exact copies of a particular spell.
Multikick abilities might be completely different from the original spell. For instance, they might print this card:
Explore 1G
Sorcery
You may put a land card from your hand onto the battlefield.
Multikick 1U: Draw a card.
On the other hand, when you Cascade, they work the same. You can pay additional costs when casting a spell and obtain the additional effects.
