As a noun, grudge is a feeling of anger or resentment that lasts for a long time. People hold a grudge.
As a verb, grudge is used often as a synonym of begrudge. However, there is a slight distinction. To grudgingly do something is to resentfully do it. The prefix be- changes grudge from an intransitive verb to a transitive verb, which means it needs to have an object receiving the action. So you must begrudge someone, but you can grudgingly do things all on your own.
In boxing two opponents can have a grudge match, which means the reason for the fight is personal. Sometimes these personal feelings can be exaggerated for publicity purposes. The term can also be used in politics.
A pretty juvenile grudge to hold, I guess. But I held them personally responsible for me not fulfilling my dream of playing for the hometown team. [Calgary Herald]
There’s nothing worse than a grudging apology. [TIME]
Now that everyone from Alcatel to ZTE has discovered that people are trying to replace phones and tablets with single hybrid devices, Apple has grudgingly decided to step into the space with the $749-and-up iPhone 6 Plus. [PC Mag]
Sargent Hopkins participated in that inaugural ride, and though the North Shore Ladies Tricycle Tours were originally intended to provide women with a sense of independence, men were begrudgingly allowed to join. [Boston Globe]
It stacks up as a grudge match between “The Finisher” and “Lionheart” — two of the Detroit area’s best mixed martial arts fighters. [Detroit Free Press]
While toss-up races in Alaska, Michigan, North Carolina and Louisiana, among others, are surely of interest, the grudge match playing out in Kentucky between Senate Minority Leader Mitch McConnell and challenger Democrat Secretary of State Alison Lundergan Grimes deserves our attention. [Fox News]