Although centrality measures tend to have positive correlations with each other, these measures all give different information so you shouldn't expect them to be high/low for the same proteins. The degree is the number of interactions that the protein makes, the closeness centrality measures how many "hops" in the network the protein is to other proteins and the betweenness centrality measures how often a protein is seen on the shortest path between other proteins. Imagine that the network represents relationships between people. If you want more people to buy you a drink, you should maximize your degree centrality i.e. have lots of friends. However, if you want to make sure that you know about every gossip making the rounds then you should maximize your betweenness centrality because having many friends doesn't help if they're isolated. Now if you're a lobbyist, you want to be able to talk to a wide range of people so you should maximize your closeness centrality.
There are other measures of centrality too that report on other properties of the nodes. Which centrality measures are relevant depends on the question you're asking about your nodes/network. For example, proteins with high degree but low closeness may form an isolated complex. Or high closeness and low betweenness could point to possible interaction "contaminants".