Their main similarities are "simple dress" and being minority sects. Early Quakers were very different from the Puritans in interpretation and living their faith. Quakers were generally pro-technology and very involved in politics unlike Amish. Puritans believed they had the one pure faith, neither Amish or Quaker have that as a core tenant of faith.

I cannot speak to issues regarding Quakers with the same familiarity as I can with issues relating to the Amish, so I shall focus mainly on them

My understanding is that there are a few Quaker denominations, if that may be the proper word, whereas the Amish have many denominations. All the different denominations have different rules regarding how strictly they follow the core ideas of simple dress, avoiding technology and so on. There is a spectrum on these topics, so more left-wing, modern Amish use cars and have phones, whereas Old Order may have a community phone and emergency radio while Nebraska Amish who are the most conservative use basically no technology.

Prior to coming to America there was one Amish church, which split into many different factions beginning in the later 1800's.

Prior to coming to America the Amish were known as some of the best farmers in Europe because of the techniques they used in farming the land

Today, depending on the group in question they may use tractors, electric ploughs or electronic milking machines. Interestingly Amish also use freezers for dairy products.

If you want a good book on the subject I highly recommend John Hostetlers Amish Society, it's the basic book on Amish history and culture