Residential proxies operated by "legitimate" providers can only open TCP connections, which more or less rules out DDoS attacks, which seem to be the most common use of botnets. (And all tcp connections typically have to go through the proxy providers datacenter first to get through NAT, which effectively limits the total amount of traffic/connections you can make)

A figleaf of "We gave ourselves permission on page 247, paragraph 3 of your user agreement"

As they are both distributed systems there will be some inherit underlying similarities, but a botnet has things like antidebug, stealth, privilege escalation, etc. They typically include attack payloads or spying features like taking screenshots, keyloggers and stealing account information. Additionally they may include further ways of spreading to other computers such as messaging people on Discord or using other vulnerabilities. They might also try and take exclusive control of the device by patching security vulnerabilities and uninstalling other malware. Well unless they sell dropper capabilities letting other people deploy malware to the machine.