A faction aims solely to advance the interests and ideological outlook of its members. It fights for total victory. It doesn’t compromise. Special-interest groups are probably the most obvious example of factionalism in action. The NRA never suggests that modest gun control would be acceptable, just as Planned Parenthood never declares that it would be content with moderate restrictions on late-term abortions. Both organizations stake out maximal positions and aim to take down anyone who deviates from them.
Parties are different — or at least they’re supposed to be in presidential systems like our own, with “winner take all” electoral rules that grant complete victory to whichever candidate (and party) manages to win a plurality of the votes in a given election.
In such systems, parties are supposed to function as coalitions of factions that join together under a broad ideological umbrella. The party seeks to obtain and hold on to political power in order to advance an agenda favored by the factional members of its coalition.