It's worth pointing out that the purpose of the placebo isn't really to test against no treatment, it's to test against the belief of receiving an effective treatment, for both the subject and the researcher. Loooong answer incoming...
Firstly, it's a lovely issue with medical trials that if you get a statistically valid comparison between a group who don't believe they're taking an effective drug and a control group that know they aren't taking anything, you'll find the placebo group has a better result. Part of it will be psychosomatic, part of it will be biased reporting of symptoms, part of it might be slight changes to routine during the trial. The end result, though, is that a placebo treatment does tend to have a slight effect in the direction intended.
IIRC, there were some less ethical studies at one point that demonstrated that if you tell a subject that their placebo drug has negative side effects, they may actually experience those effects even though the drug has no ability to cause them. Certainly there are studies in of the negative effects during studies not intended to cause them, documenting such things as withdrawal effects from being on a placebo course. In the case of negative results the term "nocebo" tends to be more appropriate.
Secondly, a placebo is important for the researchers as well. The core requirement of a double blind test is that neither the subject nor the researcher knows which treatment the subject is on. You might be able to pull off a single blind test for a comparative test but I would personally say it would be ethically problematic for a researcher to be administering an active treatment without knowing which it was. You can dodge the problem with a placebo by making the inactive treatment identical in administration to the active one.
If your test isn't double blind, there's a real risk of researchers having expectation bias skewing the data collecting and results. Unchecked this can result in tests seeming to show a treatment as being far more effective than it actually is, or as having far less serious side effects than it actually causes.
Thirdly, building on those two points and more directly answering the question, you tend to need a placebo group because that can be considered a standard reference.
So, in the first case, even assuming a blind trial, there is going to be some placebo effect going on, but having only two active groups means you can't filter out that effect. How much of the new treatment's effect was due to placebo? Would the old treatment's group have gotten better or worse results if they knew they were on a "proven" treatment? Would the new treatment's group have gotten better or worse results if they didn't think there was a chance of them being on the old treatment? You can't easily and reliably answer these questions without control groups, or if you can the results still won't be accepted as readily.
In the second case the issue is that of comparability. Even with decent correction for researcher bias you can't really get rid of the issue that the results are a comparison between two treatments. To compare to a third you'd have to try and infer it against the existing treatment's results against placebo. It'd go something like "Treatment A resulted in 10 more successful cases than Treatment B; Treatment B resulted in 20 more successful cases than placebo; Treatment C resulted in 40 more successful cases than placebo, so we thin Treatment C would be more effective than Treatment A when compared with no treatment at all". How much trust would you place in that sort of reasoning? And that's assuming you have comparisons against placebo for the other treatments... if every treatment has been measured against one or more other active treatments in studies you'll instead have some horrible network of relative efficacy to work though. Common reference points are your friend.
That's for the simple case... now imagine the more complex case. Your two groups are being tracked for a dozen or more data points, the new treatment does better than the old in some but worse in others. You have lists of side effects from both groups and there's only partial overlap there. Of course you have no control to help you remove biases and placebo effects. The resulting paper is going to end up as a straight comparison between the two treatments and a pretty high standard of objectivity and clean structuring is going to be needed to avoid a reviewer concluding that the results are subjective and unreliable. Trying to go from that paper to compare with other treatments is similarly harder.
It's easier to convince someone that your treatment is safer and more effective if you have clean studies to back it up. That's not to say comparison trials aren't useful in some cases, particularly where there's a strong ethical reason for not withholding a known treatment, but I personally would prefer to have an experiment against an inactive control group before trying to compare two active processes.
I'm also reasonably sure that tests against placebo are done with the subject's expectations being managed appropriately. Ethically the subject has to be informed that there is a chance they will be in the control group and a chance the treatment won't have the expected results. To give informed consent to the trial the subject has to understand that the treatment might not be as good as the older treatments. Thus while there is the ethical issue of effectively withholding treatment, it's being done with the consent and co-operation of the patient, hopefully with the intention of improving medicine as a whole. Having a test protocol that doesn't feature informed consent is risking some serious consequences down the line due to the much bigger ethical no-no that is forced human experimentation.
Even finding a universal cancer cure won't save a researcher from a major breach of ethics in the process of finding it.