I'm trying to understand why chemists use the mole unit instead of just counting and using SI prfixes to simplify the big numbers with units such as zetta- or yotta-molecules (yotta=10^24).
Here's what I've found so far, but it doesn't seem to be enough to make moles particularly important:
Reasons for moles:
You can easily approximate the number of atoms from the mass of a sample because the mass number of an isotope approximately equals the mass of 1 mole of atoms in grams.
It's a historical convention that would be too expensive or uncomfortable to change now.
You can use millimole and micromole instead of having to learn several more obscure SI prefixes like exa and peta. EDIT this reason is added from Jan's answer
Shorter descriptions, eg "mole of carbon" vs "yotta atom of carbon" (from matt_black's answer).
Reasons against moles:
For accurate calculations (beyond about 3 significant figures), the relationship between mass number and mass in the first reason above breaks down. So this is a risky thing to use.
It's a additional concept and set of facts that chemists have to spend effort learning but which describes human convention rather than nature.
We need extra conversion factors such as 1/mol which people often neglect and Faraday's constant used in Q = n(e-) x F  which wouldn't exist without moles.
Non-reasons for moles:
You can easily calculate the number of atoms from the mass of a sample because the atomic mass of an element equals the mass of 1 mole of atoms in grams. This relationship only exists because of the special units commonly used for atomic mass. If periodic tables listed atomic mass in grams (or perhaps yoctograms), then we could do the same calculations just as easily without moles.
In practice, we can't measure the number of molecules so we have to measure mass or volume instead and should therefore count them using a unit that's defined in terms of mass. I don't think the precise details of how a unit is defined matter for practical purposes. If you have 1 litre of an ideal gas, you still have to do calculations to find out many moles it contains just as you would to find how many molecules it contains. There is even a proposal  for SI to redefine the mole to be independent of the mass of any substance, indicating that keeping the definition isn't very important.
Chemists would make mistakes with those big numbers. They'd use "yotta" in the same way they use "mole", not doing calculations with the actual number it represents and not being at risk of other types of error. (from matt_black's answer) However it would be more complex replacing millimole with zetta because you'd have to remember that a zetta-atom is a milli-yotta-atom.
Here's a similar question but it's mixed in with the idea of measuring number rather than mass or volume - Why do people still use the mole (unit) in chemistry?
This other similar question mainly addresses the uncommonness of quantities as big as 1 mole in other areas of life - The mole is used extensively in chemistry, why not elsewhere?