what is the difference between ms-dos fat vs exfat formatting?

hello,

i would like to format my external hard drives to be mac and pc compatible. the apple store recommended MS-DOS FAT formatting while the apple support agent said to do ExFAT bec MS-DOS FAT has limitations -- you cannot have more than 32 GB storage. Is this true? Which FAT formatting should I use for a Thundebolt drive that is 1TB and an external hard drive that is 500 GB.

thanks in advance,

pia