Is it better to buy a mattress from a furniture store or from a mattress store?

It doesn't matter where you purchase. What is important is the brand of mattress and a well-trained sales associate. You could try The RoomPlace or Simmons or Tempur-Pedic.

Adding on to that, I have purchased a mattress recently from an online furniture store FabFurnish and I have exactly got what I wanted. The product and service both were Excellent!