The next iPhone's LTE speeds may be much faster. Maybe. According to 9to5Mac, Apple plans to include a new Qualcomm LTE modem in the next iPhone that's theoretically capable of delivering speeds that are twice as fast as what the iPhone 6 is capable of, with download speeds of up to 300 Mbps, up from 150 Mbps. But while the modem makes those speeds possible, they still can't happen without the right support from carriers — and in most areas of the world, you still aren't going to get anything like those speeds. That's because those speeds rely on carriers supporting LTE Advanced, which by and large hasn't rolled out.
Even if your phone isn't faster now, it could be later
LTE Advanced is already available in some areas, including Korea. AT&T has also begun supporting it in some major US cities, like New York and San Francisco, but it's unclear how widespread the service is. In practice, LTE Advanced speeds are also unlikely to match the high theoretical capabilities that the standard and these chips support. Still, it's going to make your LTE service faster to some extent, and Apple's inclusion of a new chip will future-proof the next iPhone for whenever support for this standard widely rolls out. Other smartphones, including some Galaxy phones, already include this Qualcomm chip or other support for these faster speeds.
Update: Read the iPhone 6S review.
There are other possible advantages of Apple switching over to this new LTE modem. 9to5Mac notes that it's supposed to be more power efficient, which could result in improved battery life on the next iPhone. It's also a bit smaller, and 9to5Mac suggests that Apple may make the phone's entire motherboard tinier so that a slightly larger battery can be included in the phone. However, 9to5Mac also reports that the next iPhone is unlikely to have any noticeable design changes from the current model, so the changes would likely be minor. Altogether, it's possible that those will have an impact on battery life — particularly in combination with the power saving feature being built into iOS 9 — but as Phil Schiller has suggested, Apple is pretty happy with where battery life is already at.