ACPI is newer and more cross-platform than APM from what I understand. So obviously APM is better suited for older systems than newer ones, since older systems usually don't properly support ACPI. Here's a list of pros and cons I can think of
APM cons:
APM is messy and less standardized, depending more on BIOSes than the Operating System. Since each BIOS varies, it can cause some strange, random problems system to system such as stanby crashing your system or certain devices no being compatible with your setup. The BIOS also doesn't have a clue what in the heck you are doing while you are using your computer, so powermangement can't be as "smart" as it is in ACPI.
APM Pros:
I suppose legacy support is the big thing here. Older devices should be properly supported and older computers should just work better with APM enabled. APM gives the power user control over IRQs through the BIOS, so if you are having conflicts, it is generally easy to get around them. ACPI controls much of this information. In some instances, APM has proven to be faster than ACPI because ACPI shares so many devices on one IRQ at times.
ACPI cons:
It's new, so many older devices not designed for ACPI will not work properly or at all. Those older devices that do function, usually don't support advanced ACPI features such as "sleeping".
ACPI's agressive IRQ sharing can cause your system to run more slowly than it should sometimes, but usually allows you to avoid IRQ conflicts as a whole. Also, putting your Operating System in total control of your computer can have some unexpected results such as unexplained power failures or freaky behavior.
ACPI pros:
It's the new open standard, supported by almost any modern OS. Devices and motherboards are now specifically designed for ACPI compliancy, which makes this probably the most compatible for hardware in the last few years. ACPI generally avoids IRQ conflicts where APM would have them by sharing IRQs. You also get advanced power management, allowing certain devices to "sleep", saving power. It's great for laptops and the life of your computer in general, I would imagine.
Personally, I have ACPI disabled.. Not because I don't like it, because my video card has some issues with it. If you can, I would recommend using it.