The best way to figure out whether or not a ban on "assault weapons" and "high capacity magazines" would reduce crime is to examine how such a ban worked in 1994-2004 when the US banned both. A 2004 review on the research on the effects of the "assault weapons" ban found that academic studies "did not reveal any clear impacts on gun violence outcomes" .
The University of Pennsylvania found no statistically significant evidence that either the ban on "assault weapons" or the ban on high cap magazines holding more than 10 rounds had any effect on reducing gun murders . Also, Economist John Lott studied the effects of the "assault weapons" ban and came to the same conclusion .
But then again, why would we expect gun murders to decrease in response to the "assault weapons" ban? According to the 2004 study submitted to the Department of Justice cited earlier:
"AWs [assault weapons] were used in only a small fraction of gun crimes prior to the ban: about 2% according to most studies and no more than 8%. Most of the AWs used in crime are assault pistols rather than assault rifles" Additionally, "The AW provision [of the gun control legislation] targets a relatively small number of weapons based on features that have little to do with the weapons operation, and removing those features is sufficient to make the weapons legal." The DOJ study notes that crime with "Assault weapons" decreased following the ban but states that this effect was outweighed by criminals substituting banned firearms with non-banned firearms. This isn't surprising seeing as the non-banned firearms only differ from the banned firearms in cosmetics, not in function:
"Although the ban has been successful in reducing crimes with assault weapons, any benefits from this reduction are likely to have been outweighed by steady or rising use of non-banned semi automatics with large capacity magazines, which are used more frequently in crime than assault weapons. Therefore, we cannot clearly credit the ban with any of the nation’s recent drop in gun violence." 
This is unsurprising seeing as crimes with "assault weapons" are extremely rare. It's also worth noting that ALL rifles are only used to murder around 322 people in 2012, which is an extremely small number . It's even smaller when you take into account justifiable killings by police and private citizens, which reduces it to 266 homicides in 2012. Also, most murderers don't need high cap magazines if their goal is to kill one person, never mind the fact that they could could get one illegally if they wanted.
In fact, according to a Department of Justice report listed on Dianne Feinstein's (the creator of the last ban) website , “Specific data ... suggest[s] that relatively few attacks involve more than 10 shots fired" . The report also notes, “the few available studies on shots fired show that assailants fire less than four shots on average, a number well within the 10-round magazine limit” . Thus, banning magazines over 10 bullets isn't likely to effect crime much at all, even if criminals adhered to the law. Unsurprisingly, the DOJ study reports exactly that:
"Despite a doubling of handgun LCM [large capacity magazine] prices between 1993 and 1995 and a 40% increase in rifle LCM prices from 1993 to 1994, criminal use of LCMs was rising or steady through at least the latter 1990s, based on police recovery data from four jurisdictions studied in this chapter. Post-2000 data, though more limited and inconsistent, suggest that LCM use may be dropping from peak levels of the late 1990s but provide no definitive evidence of a drop below pre-ban levels. These trends have been driven primarily by LCM handguns, which are used in crime roughly three times as often as LCM rifles. Nonetheless, there has been no consistent reduction in the use of LCM rifles either...the consistent failure to find clear evidence of a pre-post drop in LCM use across these geographically diverse locations strengthens the inference that the findings are indicative of a national pattern. " 
Overall, not only does a ban on "assault weapons" and high cap mags not do anything to deter murder, it infringes on tens of millions of gun owner's constitutionally guaranteed rights.
EDIT: The study also finds that the percent of firearms collected by police that are considered "assault weapons" decreased following the ban. However, the study also notes, "it is worth noting the ban has not completely eliminated the use of AWs, and, despite large relative reductions, the share of gun crimes involving AWs is similar to that before the ban. Based on year 2000 or more recent data, the most common AWs continue to be used in up to 1.7% of gun crimes." 
Later in the study the authors write: "there has been no discernible reduction in the lethality and
injuriousness of gun violence, based on indicators like the percentage of gun crimes
resulting in death or the share of gunfire incidents resulting in injury, as we might have
expected had the ban reduced crimes with both AWs and LCMs." This kind of contradicts their earlier statement that gun crime with AWs was reduced after the ban.