This is a topic that has been discussed many times, so I'd suggest doing a search if you haven't already.

In the meantime, I'll try and share my opinion, based on what I've been able to research regarding terminal ballistics.

TL;DR: HP has a higher probability of disrupting tissue and a lower probability of overpenetrating than RN, but nothing can replace shot placement as the most important factor.

For my reasoning...

Two main issues: Wounding and overpenetration.

Wounding

Simply put, HP has more potential to damage tissue than RN. Tissue is elastic. With RN, tissue is going to stretch until it hits its elastic limit before the bullet will penetrate. After the bullet passes through, the elastic nature of the punctured tissue will cause it to return to its normal state, which means the bullet hole will actually be smaller than the caliber of the bullet that caused it. In some cases, it may even close up, minimizing trauma and/or blood loss.

The rounded profile is also more likely to ricochet off bones instead of breaking them, unless the bullet hits the bone dead-on. Similarly, it may be more likely to push aside blood vessels rather than cut or tear them, further reducing its wounding ability.

With a HP, the expanding bullet will actually crush more tissue, and be more likely to cut a wound channel larger than its caliber. It is also more likely to dig into bone and damage it, even if the hit isn't perfectly centered. Similarly, it's more likely to cut or tear blood vessels, leading to more blood loss. Unless you get a CNS hit, blood loss is probably going to be the primary physiological cause of an attacker stopping after being shot (psychological stops are a factor, but not something I'd count on).

Even if a HP bullet doesn't expand, it still has a relatively sharp edge with a cookie-cutter effect, so it still has a better potential for disrupting tissue than RN bullets.

Some will argue that this doesn't happen, or that HP is no better than RN when it comes to effectiveness. Even if that is true, HP has a documented advantage over RN in one aspect...

Overpenetration

Despite what some say, overpenetration is a very real concern. There've been several documented instances of RN bullets exiting people and injuring others. NYPD changed from FMJ to JHP because of numerous instances of FMJ rounds exiting and injuring not only bystanders, but also other officers. Two instances of .45ACP FMJ overpenetrating and causing injuries comes to mind. In one, an officer fired at a suspect. The .45ACP FMJ bullet exited the suspect and struck a second officer, who was coming to the first officer's aid, in the abdomen below his vest, causing a life-threatening injury. In the other, a SWAT officer shot a hostage taker with .45ACP FMJ and the bullet exited, injuring one of the hostages.

Can HP exit? Sure, but 1) it's not as likely, and 2) it's not as dangerous because of the velocity loss caused by expansion and slowing down as it travels through the attacker's body. I've read about one shooting, I believe with 180gr .40S&W, in which the round exited a suspect's body, but was stopped by the suspect's T-shirt. Another one involved a NYPD officer firing a 124gr GDHP +P at a suspect. The round exited, with textbook expansion, and was stopped by the suspect's jacket. One incident involved .357Sig exiting, but falling on the ground about 10ft behind the suspect, indicating a significant loss of velocity and a much reduced likelihood of causing an injury.

Penetration is important. The deeper a bullet penetrates, the more tissue it can disrupt. But we also want to reduce the probability of overpenetration. Fortunately, there are plenty of HP options that strike a good balance between adequate penetration and expansion to minimize that risk.

Everything we do with regard to self defense, and, in a way, life in general, is all based on probabilities. Whatever hardware (guns, bullets, etc.) and software (shooting techniques, tactics, etc.) we choose to use, we make those choices on the basis that it will improve our odds in a self defense encounter. The way I see it, I want to maximize those odds in my favor, and choosing HP over RN is one factor we have control over. Obviously, people can choose what they want, for whatever reason they want, but the evidence I've seen suggests HP will usually be the better choice.

Unfortunately, with the ammo market being the way it is, good HP ammo may be difficult, if not impossible, to find, so we may just have to do the best we can with what we have or what we can get. Regardless of bullet type, shot placement is still the most important factor in stopping a violent attacker.

Just my opinion.

And sorry for the novel.