Tonality is not the same as value or tone although it helps to explain value or tonal relationships. While value refers to the relative lightness or darkness of things independent of color (as in a black and white photograph), tonality has to do with the way colors unify.
When Monet said it is the "surrounding atmosphere which gives subjects their true value" he was referring to tonality or quality of light (atmosphere) in which a subject exists. Tonality is the quality of light that bathes everything.
Think of it this way: suppose it were midnight in a dark room and I turned on a soft green light. Everything would be a little greenish. If I changed the light to yellow, everything would a little yellowish, and so on. The problem arises when the light is 'normal' daylight because we generally don't see the tonality. It is as though we were like fish that don't know they are in water. In fact, we might better understand tonality if we, indeed, think of atmosphere as a medium like water in which we live. Thus the sky isn't a curtain behind the mountains. We are in the sky, under it -- living, acting, and moving within it.
How to See Tonality
Absent tonality, our paintings are apt to appear simply as a collection of separate things. It would be terribly difficult to achieve the kind of harmony or unity that tonality provides by simply trying to make the colors of the separate things work. The trick, of course, is to see tonality. To do this, it helps to understand that it is impossible to know the color of the thing except as it is mediated by the surrounding "atmosphere".
In the two still life paintings shown here, the compositions vary but the apples, leaves, cloth, and tabletop are identical. However, the one with cooler tonality was done in north light while the one with a warmer tonality was under an incandescent light. Tone painters (George Inness and Russell Chatham are examples) seemingly point to the beauty of tonality.
Don't think house, water, flesh; rather, look through the little vales of atmosphere and enjoy seeing the "ishes" -- bluish, greenish, reddish, and get the thing through the color. Squint and compare so you can properly relate both color and value. Then you will get the tonality. Your paintings will have more mood, and more of you in them.