Nazarite Vow

Perhaps because the Nazerite vow was usually for a limited period of time during which the man or woman separated themselves for devotion unto God. According to Num. 6:18, the long hair or beard was to be cut off when the period of separation was over and put it in the fire as part of a fellowship offering in a special ceremony at the tabernacle.

So it would appear that the "extra hair" grown during this time of the vow(separation) was considered to be "dedicated" to the offering of the Lord. (vs. 9)

Acts 21:22-24 describes four such men who taken the vow and would have had long hair to offer as a sacrifice described in verse 26.
The fact that this vow required this type of abandonment of normal grooming was apparantly designed to show one's complete devotion to the Lord and a temporary casting aside of adornment. So Paul's words in 1 Cor. 11:14 speak to men who were not having long hair as a sign of devotion or separation unto the Lord, but as it being grown for their own glory. And since the man's glory is not be found in him, but in his Head, which is Christ, then his long hair would be seen as competing for the glory that is only to be reserved for Jesus Christ and His Head which is God.
Which is why, incidently, I believe that this passage teaches that a woman should cover her glory, which is her hair(ie. some advocate it must be long, whatever length that means, I'm not sure) when she is in the public worship of the gathered believers, whereas a man should leave his hair uncovered since he is the glory of God. The woman covers her glory so that the only glory that is visible is that man's which is Christ, our Head. And the covering shows that she is under the authority of her head.
But I don't want to derail the OP by debating head coverings for women. Although that's always an interesting topic.

 

Somebody mention long hair?

 

Exactly. A similar humiliating sign of submission to God was the wearing of sackcloth and ashes in public.