If your server gets hacked you don't have any customer info stored on it. All you have is a stripe token.
ONE: You don't use stripe and save all you customer info locally and procces it some other way
TWO: You use stripe and never store any CC info on your server.
THEN: Your server is hacked. In ONE, you lose customer CC info because it is stored on your server. In TWO, you do NOT. Sure they can redirect you to a fake payment page, but that will get noticed quickly and damage will be limited to "phished" users not your entire user base.
While not over the network, the user enters his credit card into the form ON YOUR WEBSITE that POSTs it on to stripe. This means your site has full access to the credit card info, choosing whether or not to store is up to you (or in case you get hacked, up to the hacker).
The attacker can replace the iframe with his own and proxy it, or perhaps simply display an error and ask the user to re-insert his payment info into the actual stripe form.
Flow 1: User visits shop.com, selects some items, hits checkout, enters his credit card details into a form on shop.com, hits submit, gets emailed a receipt.
Flow 2: User visits shop.com, selects some items, hits checkout, is redirected to a form on processor.com, enters his card details, hits submit, gets email a receipt.
Now flow 1 and flow 2 could be the same internally if the form in flow 1 was actually from processor.com, just embedded into an iframe displayed on shop.com. But the key is that the user has no idea if it is or not.
The form in frame 1 could be in an iframe from anywhere, or it might actually be embedded directly into the page and posting, again, to anywhere. You just have to take it on faith that shop.com hasn't been compromised, and isn't now sending your details to malware.com.
In flow 2, you only enter your details into a form on processor.com; you no longer need to worry if shop.com has been compromised, you just need to verify that you're actually at processor.com, and not proccesor.com or whatever.
I'm not a big believer in the usefulness of schemes like PCI DSS anyway, but to the extent that they might be useful, it's as a checklist of good practices to follow when things are working properly. In that case, clearly there is a significant distinction between sending sensitive data over your own network to be stored on your own servers and sending sensitive data only to a trusted service provider that is better equipped to deal with it securely.