Whenever the Doctor regenerates, his body and personality change. The tenth and eleventh doctors have different personalities. So when Mel in "Let's Kill Hitler" regenerates into River Song her personality is the same. Why is that?
Three possibilities I can think of based on the reboot series
1. Melody/River is part human
In the control room, Dorium hacks into Kovarian's files and finds scans of Melody's DNA, which contain traces of Time Lord DNA. Vastra wonders where Melody was conceived, knowing that Time Lords became what they were through exposure to the time vortex and the Untempered Schism. At first, the Doctor does not want to think about the event in question and goes off on an awkwardly spun tangent to avoid the touchy subject. However, his avoidance of the question leads him straight to the answer.
Figuring out his huge blunder, he stops dead in the middle of his yammering, leaving Vastra to prompt him to finish his sentence and Dorium inquiring, "Mmm...?". The Doctor remembers that the first time Amy and Rory were together on the TARDIS in this version of reality was on their wedding night. Knowing he left the couple alone and this is his fault, the Doctor gawks with his mouth agape and a giant blank stare. Vastra deduces that Kovarian wants a part-Time Lord to turn her into a weapon.
Source: tardis.wikia.com: A Good Man Goes to War (TV story) > Plot
Since River was conceived by two humans she would have normally been human, but as Vastra says, Time Lords became what they were though exposure to the time vortex and the Untempered Schism to which we see back in The Sound of Drums where The Doctor tells Martha and Jack of The Master's up-bring and how young Gallifreyan are taken to the Untempered Schism at a certain age.
As the Doctor says in response to Vastra's comment, it took ages for Time Lords to become what they are now. However, it's possible that the process would be sped up if the child was born in the Time Vortex.
DOCTOR: But she's human. She's Amy and Rory's daughter.
VASTRA: You've told me about your people. They became what they did through prolonged exposure to the time vortex. The Untempered Schism.
DOCTOR: Over billions of years. It didn't just happen.
VASTRA: So how close is she? Could she even regenerate?
DOCTOR: No, no. I don't think so.
VASTRA: You don't sound so sure.
DOCTOR: Because I don't understand how this happened.
VASTRA: Which leads me to ask when did it happen?
VASTRA: I am trying to be delicate. I know how you can blush. When did this baby begin?
DOCTOR: Oh, you mean
DOCTOR: Well, how would I know? That's all human-y, private stuff. It just sort of goes on. They don't put up a balloon, or anything.
VASTRA: But could the child have begun on the Tardis in flight, in the vortex.
DOCTOR: No! No! Impossible! It's all running about, sexy fish vampires and blowing up stuff. And Rory wasn't even there at the beginning. Then he was dead, then he didn't exist, then he was plastic. Then I had to reboot the whole universe. Long story. So, technically the first time they were on the Tardis together in this version of reality, was on their w
VASTRA: On their what?
DOCTOR: On their wedding night.
It could be that the human part of Melody/River allows her to retain more of her personality
2. Melody/River's Personality was Programmed to remain the same
Remember that the goal by The Silence was to use Melody/River to kill The Doctor.
AMY: You are not a psychopath. Why would she be a psychopath?
RIVER: Oh, Mummy, Mummy, pay attention. I was trained and conditioned for one purpose. I was born to kill the Doctor.
DOCTOR: Demons Run, remember? This is what they were building. My bespoke psychopath.
RIVER: I'm all yours, sweetie.
(She kisses him lightly.)
The Silence wanted Melody/River because she was part Time Lord, otherwise any child would have done and if the plan was to use a child the Doctor wouldn't want to harm (because they are the child of a close friend) there's still a bunch of his allies who they could have pulled from the timeline.
Given also that River was taken from the Luna University in 5123 to be put in Lake Silencio in 2011 it would suggest that while working on her in 1969 they needed her to still be around in 2011. I suspect they wanted her to regenerate as well, as shown at the end of Day of the Moon where the child Melody regenerated some time after escaping and while saying she was dying.
TRAMP: Are you okay? Little girl, are you okay?
GIRL: It's all right. It's quite all right. I'm dying. But I can fix that. It's easy, really. See?
The Silence wanted a part Time Lord child that could regenerate, developed a poison to stop The Doctor regenerating, and then set a trap to kill him mid regeneration to permanently kill him. I would think that The Silence did their research on Time Lord regeneration and so as part of Melody's conditioning to kill her parent's best friend, they took steps to make sure that she would still be a psychopath after regeneration.
3. It was controlled regeneration
The regeneration can be controlled as shown by The Doctor himself
- In Journey's End the Tenth Doctor's has enough control to use his old hand he got from Jack Harkness back in Utopia to keep his current form.
- The Eleventh Doctor chose his next form because of the event of The Fires of Pompeii when he was with Donna Noble to remember a promise he made (to himself)
- The Twelfth Doctor, even after getting blown up. tried to hold off regenerating at the end of The Doctor Falls and in Twice Upon A Time (NOTE: I missed the special but I assume it was Peter Capaldi for a good part of it).
We also suspect there was a degree of control when Melody regenerated.
AMY: You're Melody?
RORY: But if she's Melody, that means that she's also
MELS: Shut up, Dad. I'm focusing on a dress size.
(And Mels transforms into -)
RIVER: Oh! Oh! Oh! Whoa! Right, let's see, then. Ooo, it's all going on down there, isn't it? The hair! Oh, the hair. It just doesn't stop, does it? Look at that. Everything changes. Oh, but I love it. I love it! I'm all sort of mature.
We see here that Melody was trying to focus on a dress size but I admit, we don't know if it worked (though the fact that she's focusing on a dress size means it could have happened before)