They are two entirely different statements, and, while there are cases where both apply, by no means are they equivalent.

They're obviously not completely equivalent, but in cases where your measurements form some Gaussian (or similar) distribution, which is very common, the you have the choice of saying things like (to use the water-purifying example), "we're 85% confident it's at least 99.97% pure", "we're 97.7% confident it's at least 99.3% pure", "We're 99.9% confident it's at least 98.5% pure", etc., etc., each of which represents a different part of the curve. Now obviously the most complete answer here would be to say "our data are decribed by a Gaussian of mean X and st. dev. Y", but people don't frequently do that in informal contexts, so how do you reduce it to one claim with one confidence?

In the relationship context, "better off" tends to come in large, uncertain chunks, so what rhollerith said is reasonable, what you suggested he might say seems very unlikely to be the case.

Would you go into why that is? It doesn't seem intuitive to me at all. Why shouldn't a relationship improve your life by just a small amount?