Cromwell was a Puritan, and Puritan were scriptural literal-ists. They found no scriptural justification for the celebration of Christmas. They associated it with paganism, and residual Papist idolatry.
To be fair Christmas traditions in seventeenth century Britain were pretty boisterous..
The Puritans believed in plain dress, no singing (outside of prayer), no dancing, one can see why they didn't like many Christmas traditions.
Christmas in 17th-Century England and Virginia
In preparation for the season, many towns designated a Lord of Misrule, the “grand captain of all mischief,” who, with 20 or more chosen “lusty guts,” decked themselves in yellow and green scarves, ribbons, laces, rings and jewels, and processed through the town on Christmas Day. In the late 16th century, Philip Stubbes, of puritanical leanings, related how this “heathen company” marched “towards the church and churchyard, their pipers piping, their drummers thundering, their stumps dancing, their bells jingling, their handkerchiefs swinging about their heads like madmen, their hobby-horses and other monsters skirmishing amongst the rout.” Stubbes and others argued for an end to the licentiousness and revelry often associated with the Lord of Misrule and his mummers. This custom, however, was so ingrained in the minds of Englishmen of all classes, that even with the rise of Puritans to political power in the 1640s, attempts at controlling Christmas merriment often failed.
Puritans preferred few holidays, and when they did justify work stoppages they preferred days of humiliation and prayer to Catholic holiday based around pagan traditions.