The LHC@Home was initially launched[1] to simulate the LHC detector and optimize machine parameters before the LHC went live. It received little to no funding for years, however without LHC@Home a lot of the work would very likely not have been done on existing supercomputers[2]:
"Some 1.5x10^5 users with more than 3x10^5 PCs have been active LHC@Home volunteers since its launch. This has provided significant computing power for accelerator physics studies, for which there was no equivalent capacity available in the regular CERN computing clusters. Volunteers contributing to SixTrack have delivered a sustained processing capacity of more than 45 TeraFlops."
The project has been a huge success and the studies done using it, such as this one[3], has helped to extract much more physics than anticipated[4].
So at least there are some @Home projects that have a very real impact.
"Some 1.5x10^5 users with more than 3x10^5 PCs have been active LHC@Home volunteers since its launch. This has provided significant computing power for accelerator physics studies, for which there was no equivalent capacity available in the regular CERN computing clusters. Volunteers contributing to SixTrack have delivered a sustained processing capacity of more than 45 TeraFlops."
The project has been a huge success and the studies done using it, such as this one[3], has helped to extract much more physics than anticipated[4].
So at least there are some @Home projects that have a very real impact.
[1]: https://lhcathome.web.cern.ch/about/history
[2]: http://cds.cern.ch/record/2301793?ln=en
[3]: http://cds.cern.ch/record/972345?ln=en
[4]: https://www.physicsforums.com/threads/lhc-ended-2016-proton-...