My Baofeng UV-5R came with a shorter 6.5” antenna and a “high gain” 15.3” RH-771 antenna. In your experience, how much of a difference in range does the use of the longer antenna achieve?
This is a harder question to answer than you think for a number of reasons. I'll start off with the ideal answer:
The Diamond RH-771 claims that it has 2.15 dBi gain at 70cm (or 0 dBd). Most rubber ducks are significantly worse than quarter wave antennas, and often exhibit negative gain. A short antenna like the SRH-701 has no gain (or -2.15 dBd). The difference is 2.15 dB, so you'd expect a radio with the larger antenna to be about 1.64 times more powerful on transmit and receive.
In other words, a 3 watt radio using the longer antenna will go about the same distance as a 5 watt radio using the shorter antenna. One common rule of thumb is about 1 km for every half watt, so you'd expect to go about 4 km farther with the long antenna than the short antenna.
What I just said is a lie.
Here are all the things I ignored when doing those calculations.
The radio horizon. The distance your signal can travel depends to a large extent on the height the antenna is from the ground. Antennas that are up higher will go further than antennas that are lower. If you pick 2 m as the average height for a human, the radio horizon is about 6 km. If you're transmitting to another person who's also about 2m tall, it doesn't matter how powerful your radio is - it won't be able to go further than (6+6) km around the curvature of the earth. Here's a calculator. That's why repeaters are on tall things - they increase the effective range of handhelds by making the RF horizon distance larger.
Radio waves bounce off things. I just told you that you're not going to be able to hear anyone more than 12 km away going handheld-to-handheld. So how come you can sometimes go greater distances? Radio waves don't just follow a circle around the earth. They will sometimes bounce off planes flying by, water towers, buildings - effectively increasing the distance a signal can travel.
You don't have a Diamond RH-771. There are a lot of cheaper antennas out there. Some are good, some are not - and the one you got with the Baofeng is, I'm afraid, probably not quite up to Diamond in terms of quality. I'd bet you have a clone of the Nagoya 771. That doesn't mean it's not good, but it means that my assumption that it exhibits 2.15 dBi gain is probably not true.
Radio sensitivity. The Baofeng UV-5R is in a class of radios that is pretty inexpensive. That's not a bad thing - it brings people like you into the hobby - but one of the downsides of inexpensive radios is that they don't perform well when there are a lot of other RF signals around, such as you might encounter in a city. Think about trying to hear someone whispering to you (another handheld) when you have a freight train going by (multiple high-powered transmitters). When you have a lousy antenna, it does a poor job of pulling in both signals. A better antenna amplifies both the whisper and the freight train, which means your poor radio is deafened by the ambient noise and can't pull out the weak signal from the handheld. More expensive radios have better filtering, and will be more effective in high-RF environments. If you're out in the country, you're likely to hear farther with the longer antenna because your radio will be more sensitive.
Antenna polarization. If your antenna and the other radio's antenna are both verticals pointing upward, they're polarized the same way. If one of them is pointing sideways and the other points up, they're cross-polarized. In practice this is like losing 15-40 dB which is much more than any gain you'll get from the antenna.
Now that I've said all this, you can go further than you think you can. Stations with Baofeng radios have worked the ISS (that's at least 408 km away, straight up) using a vertical antenna. There are no buildings in the way, and they're doing it from areas that are RF-quiet.
In general, a longer dual-band antenna will perform better than a shorter dual-band antenna. How high up you are, how well your radio filters other noisy transmitters, transmitting and receiving antenna polarization, and how many things are between you and the other station all will influence how far you can hear and be heard.