We can’t predict how your company or peers will react, nor can we see the potential downstream consequences of doing this to your employer.
It’s possible that they won’t care and it won’t matter. Or it’s possible that your peers and managers will take it personally and forever think of you as the unprofessional guy who abandoned their project with less than 2 weeks’ notice. Those negative perceptions could influence future opportunities as your peers move to other companies and are eventually asked what kind of employee you are. Or maybe none of your peers are ever asked for back channel references.
We can’t know the outcome, but know that you’re gambling with your reputation a bit. Good reputations are hard to build but easy to destroy. Be careful.
> In my specific situation, I want to start the new job fast but I also want a week off. Giving two weeks to my current job would also coincide with vacation I had previously planned.
You can’t have it all. I’d suggest telling your new job you’re available in 3 weeks and then splitting your 2 weeks’ notice around the vacation. There’s a chance that your employer could cut you loose early, in which case you go back to the new employer and ask if you can move your start date up.