Historically, it was thought that the surgical approach was significantly superior to the non-surgical approach when treating achilles tendon ruptures. The key advantage of surgical treatment lay in significantly lower re-rupture rates. On average, re-rupture rates after non-surgical treatment were around 20 percent (with a high of 40 percent!) and re-rupture rates from surgical treatment were around 5 percent or less in most studies.
However, since 2008 or so, a lot of new studies have concluded that non-surgical treatment is just as effective as surgical treatment and results in just slightly higher re-rupture rates. The key factors in a successful outcome and low re-rupture potential from non-surgical treatment are functional bracing and early weight bearing and mobilization. See studies in support of non-surgical treatment of achilles tendon ruptures for more on that.
Early weight bearing was largely discouraged until recently so on this page, I will only list newer studies in support of surgical treatment and ignore older studies.
2013, Sweden — This is not a study, but a superb thesis. It includes a literature review section, as well as analysis of four unique studies. The results of the studies suggest that surgical outcomes are generally slightly favorable in comparison to non-surgical outcomes in terms of modestly better final functional outcomes and lower re-rupture rates.