AAS is cheaper but you can only quantitate 1 element at the time (>1 ppm). ICP-MS uses 'very pure' and expensive reference standards.
It depends a lot on your sample matrices, LOQ, extraction techniques. If you have a lot of water samples needing Ca and Mg levels I would use AAS as my work-horse and ICP for 'special' samples.
Obviously ICP-MS is more sensitive and simultaneous but very expensive. It depend what you want to do. Using ICP-MS for analysis needs proficient operator. Operating AAS and processing it's data is much easier.
None! Only the element identification and the concentration is the same. The ICP uses an MS (mass spectrometric detector) while AAS uses a UV-Visible spectrophotometer as a detector. The ICP (plasma) has a higher flame temperature so ALL the atoms are in emission while the AAS has a much cooler flame and thus atoms can either emit or absorb radiation.
The ICP-MS is 1000 times better then AAS as the sensitivity is 1000 times more. It also has more throughput then AAS and can perform multiple element in a single run. The application of AAS is limited to few elements but ICP-MS can analyze maximum elements in the periodic table.
You can order this ultrapure ICP standard from any chemical supply house (it was typically used for ppb concentations of harmful elements in environmental samples). Just compare it to the standard you used and it will make your sample results comparable.