These two mattresses are very similar as both are foam, 10 inches in profile, and both are at the same price point of $600 for a queen mattress, however, there are some differences.
The Tuft & Needle mattress is made in the US while the Quatro Sleep Mattress is made in China. Tuft & Needle mattress has two layers foam on top of base foam while the Quatro Sleep Mattress has a three-layer construction with two layers of memory foam on top of the base foam.