Both beam energy and luminosity are important for succesful particle accelerator experiments. The LHC's nominal design is for 7 TeV beams and 1E34 cm^-2 sec^-1 luminosity, while the SSC's nominal design was for 20 TeV beams and 1E33 cm^-2 sec^-1 luminosity. The LHC has 10x greater luminosity while the SSC would have had almost 3x greater energy. If the SSC was alive and both machines were operating at nominal levels, which would have the greater advantage for discovery of: i. the Higgs boson, ii. supersymmetry, iii. extra dimensions? I'm asking on the assumption that all three of these exist.
Dear Michael, with the energies and luminosities you described (which are not necessarily those that may be quickly achieved by either machine, and the LHC is still not there), the SSC would be a more potent machine to discover all the things you mentioned, especially the physics associated with the very heavy scales - a few TeV (which is surely extra dimensions, if they exist, and maybe even SUSY) - that the LHC may be unable to reach.
The new particles near the upper bound of a collider's energy reach become extremely rare - so the LHC is producing e.g. a top quark many times a second while it used to be produced once a month at a lower energy but comparable luminosity by the Tevatron.
But even if the luminosity of a higher-energy machine is smaller, it may still produce many more particles even if they're lighter because a more energetic collision contains more particles in the final state. So for example, despite the fact that the LHC has only collided 0.3/fb in each detector, it has beaten the Tevatron with its 10/fb in pretty much everything - and in some "disciplines", the LHC advantage has become overwhelming. Of course, examples of the latter include supersymmetry - the exclusion limits by the LHC are far more far-reaching than those by the Tevatron and the Tevatron couldn't even match the current LHC results even if it were running for another decade.
There remain some special features, such as the top-antitop asymmetry, that are harder to be seen on the LHC than on the Tevatron because the latter is a proton-antiproton collider.
Despite the advantage of the SSC, it would be a kind of a brute-force, expensive machine. The LHC is doing many things in a smarter, cheaper, and technologically more advanced way. The names may be confusing. The SSC was "super", like in "superconducting". Of course, the magnets in the LHC are also superconducting and technologically more advanced, in fact, than the SSC plans.
Beacuse of its higher energy the SSC would have been able to search for SUSY and other BSM exotics with higher masses so it would have been superior in that respect, but it is not so clear that it would have been superior for finding the Higgs.
There are other factors apart from energy and luminosity that were key to the LHC discovering the Higgs before the Tevatron. The detector energy resolution was arguably the most important factor. The Tevatron had energy resolutions around 5GeV whereas the LHC detectors had 1 GeV resolution for diphotons and 4leptons. This means that the LHC could measure the energy of the particles much more accurately.
When searching for the Higgs there is a lot of background in these channels and the higher energy only makes the background stronger. It is the higher energy resolution that makes it easier to extract the signal from the noise. Note that in other channels such as the dominant bb decay the LHC has not yet matched the tevatron despite the higher energy and total luminosity. That is because the LHC does not have much better resolution than the Tevatron for jets and the higher energy only produces more background.
It is not clear what all the specs of the SSC detectors would have been but it is likely that they would have been initially similar to the Tevatron's detectors of that time. In that case with the lower luminosity and higher energy it would have been difficult for the SSC to do better for finding the Higgs than the Tevatron did. It is possible that the detectors would have been upgraded later to be more like the ones at the LHC but that might not have happened any sooner.
There are other factors that could have been problematical for the SSC. One is that they were considering a proton-antiproton collider like the Tevatron. This means that they would have had to build up a store of antiprotons before each fill and if the beam is dumped it could take a day to generate a new store. This was a limiting factor for the Tevatron. From experience with the LHC we know that holding stable beams at higher energies is not easy. The design of the LHC included advanced magnets that could help with beam stabilization and the SSC would have needed those too. But the good thing about the LHC was that it was a proton-proton collider so when the beam was lost they could produce a new one relatively quickly.
There are other factors that might have been a problem for the SSC due to less advanced technology of the time and the bigger jump in energy being attempted. For example they would have had less computing power and data storage at the time making it difficult to gather enough events. They might also have suffered badly from problems such as e-clouding and UFOs that have been discovered at the LHC and get worse with increasing energy.
In conclusion it is certain that the SSC would have been better for BSM physics assuming they reached its design spec but whether it could beat the LHC to the Higgs would depend on some design and technological considerations that are uncertain given that it did not reach completion. It seems likely that this would have been problematical given that the Higgs turned out to be in the most difficult mass range to search where seeing diphoton and 4lepton events clearly was crucial.