Strength and Toughness

Image source: Materials Group - University of Cambridge
We need the separate words "strength" and "toughness" because sometimes materials with high toughness and high strength are very different, like rubber and ceramic. Let's say you are designing a chair so that can support a person of a certain weight. Why not build it out of glass right? As long as the strength is high enough, it will support that person, you reason. The problem is if a person of weight above the design weight sits on the chair, it will shatter immediately because of its low toughness. Whereas a chair made out of polymers (plastic) may have the same strength as the glass but also have some toughness so that it would deform a bit before breaking. Final example: steel and ceramic both have high strength but very different toughness, so you need both words to distinguish their behaviors.
Strength and Hardness
While they do seem to be correlated, they are not exactly the same:

Image source: "Predicting the compressive and tensile strength of rocks from indentation hardness index" by Kahraman, Fener, and Kozman. Journal of the Southern African Institute of Mining and Metallurgy. 2012.
The figure above contains data on rocks but similar correlations are found in metals as well. As with strength and toughness, two materials may have the same strength but different hardness. Maybe people keep dragging sharp plows along your steel bridge and they are slowly scraping material away, so independent of how heavy those plows are (which determines the strength required) you decide to use a harder material.