Bigger stakes, too. In the show, they were just another ship (one of 12 or so...) exploring the outer reaches on behalf of an otherwise unseen agency. In the movies (including TNG and Kelvinverse), they're the greatest heroes of all time, saving Earth itself five or six different times out of 13 films (no Earth in Insurrection and Beyond, both considered "episodic" by fans).
I wouldn't say a threat to Earth is necessarily "bigger," just closer to home. I mean, if Khan or Kruge had gotten Genesis, that could've endangered dozens or hundreds of worlds, not just one. If the God Entity had escaped from Sha Ka Ree, much of the galaxy could've been endangered, or at least that was implied, though not really effectively shown. A renewed state of war with a dying and desperate Klingon Empire could've devastated much of the Federation. The Borg are a galactic-scale threat whether they jeopardize Earth or not. And Shinzon's attack on Earth was meant to be just the opening salvo in a larger war of conquest, while Nero's was the second attack in his campaign to wipe out all the major worlds of the Federation. Admiral Marcus's scheme to engineer a war would've devastated many worlds, and Krall intended to lead the Swarm on a war of conquest against the Federation at large.
So if anything, I'd say the movies where Earth was directly threatened tended to have
smaller stakes, or else had stakes that would still have been quite large even without Earth being involved. V'Ger and the Whale Probe didn't seem to have any specific hostile agenda beyond Earth, and it was just a starting point for Shinzon and Nero. The only movie in which the attack on Earth raised the stakes in and of itself (beyond the general disruption to the Federation if it lost its capital) was
First Contact, because the time-travel attack on Earth's past would have wiped out the whole Federation and probably much of the rest of the galaxy.
I believe there was also some general thought that they didn't want to get into predicting what kind of socio-economic system won the Cold War. Was Future Earth capitalist, communist, socialist, or something altogether different? As I recall, the original writers guide's advised getting too specific about what Future Earth was like to avoid those kind of predictions.
Clearly, this policy fell by the wayside when it came to the later movies and TV series.
It was also to avoid the risk of making predictions that would look outdated or silly later on, like predicting that a technology would develop much slower than in real life (like handheld communication boxes) or much faster than in real life (like manned Saturn probes and cryogenic sleeper ships). That's why TOS tried to be vague about just how far in the future it was -- another policy that fell by the wayside later on.
