Contrary to the suggestive nature of the question, the answer already produced by Ergative Man, and a couple of the comments, there is an important difference between the constituent and the phrase. A phrase is a sub-type of constituent, which means every phrase is a constituent but many constituents are not phrases.
For orientation, I think many syntacticians would agree with the following definition of the constituent unit when considering tree analyses:
Constituent: A node N plus everything that N dominates.
With this definition in mind, consider the next tree analysis, which I have taken from the Wikipedia article on the constituent unit cited in the question:

While one can certainly disagree with the structural analysis shown here, its validity is taken for granted for the sake of illustration. According to the definition above and this tree, there are nine distinct constituents present:
Drunks,
could,
put,
off,
the,
customers,
the customers,
put off the customers, and
could put of the customers.
But according to the labels given in the tree, there are only three distinct phrases present:
the customers,
put off the customers, and
could put off the customers.
The difference, then, lies in particular with what one considers a phrase to be. I think many syntacticians would disagree with the tree analysis above concerning the subject Drunks; they would view it as a phrase because it could take dependents but simply does not do so in this particular case. Few syntacticians would ever choose to view a particle like off or the definite article the as phrases, though, because these words rarely if ever take dependents.
Consider the following tree analysis of the same sentence next; it is the dependency grammar (DG) analysis and is also taken from the Wikipedia article that the question cites:

Applying the definition of the constituent above to this analysis, the following words and word combinations are constituents; there are just five of them:
Drunks,
off,
the,
the customers, and
put off the customers.
While the labels used do not identify phrases, I think some DG people would define the phrase as two or more words that form a complete subtree. On this definition, the tree shows just two phrases:
the customers, and
put off the customers.
The examples just discussed illustrate that a given phrase is always a constituent, but many constituents are not phrases. The constituent is a more inclusive unit than the phrase. I think most grammarians would agree about this. When one looks further, though, to which constituents should be construed as phrases, opinions vary, sometimes drastically so.