Indeed, Botanicula is very different than Machinarium and so to, for that matter, is Samorost 3. So if you're looking for a "continuation of Machinarium" then you will be disappointed with Samorost 3.
But you can play Samorost 1 and the first "half" of Samorost 2 for free and make your mind up from there. You'll find the links in the "Games" tab up top.
As for your question about whether you need to play Samorost 1 & 2 first, just look in the
Samorost 3 FAQ (the short answer is, no)