The previous answers are accurate. "Count on" and "count with" are definitely different. That said, they are related!
Imagine I grow trees I want to post a sign that says "Apple Trees for Sale".
I ask my neighbor to bring me some paint. He says "You can count on me."
So I can "count" that first step as good as done, then I can count/figure what the next steps are and how I'll get them done--as if I am counting on my fingers "1, 2, 3, 4" the steps and how I'll get them done.
So I can have an idea of how I'll do all this before I actually begin.
I've seen the phrase interpreted as "you can trust me". This is close, but too simple. The phrase means "you can depend on me to do this specific thing". Related is "you can trust me to do the right thing in general". I'm pretty sure the first definition is how the phrase started.
There's a reason it's "count on me" instead of "milk on me", "walk on me", "fly on me". The phrase has a logic, sense and history to it. Many words are like this and can be rewarding+helpful to learn about.
--A similar phrase is "counting on". As in, "I was counting on that pay raise in order to go vacation". This means that I was depending on the extra money I figured I would get.