brew services (services in general) will restart automatically after rebooting; the other (mysql.server start) will not. That is the difference. Executing the command with no options should yield,
tigger2:Downloads efrisch$ brew services brew services [-v|--verbose] [list | run | start | stop | restart | cleanup] [...] Easily start and stop formulae via launchctl. With -v or --verbose, print more detail. Integrates Homebrew formulae with macOS' launchctl manager. Services can be added to either /Library/LaunchDaemons or ~/Library/LaunchAgents. Basically, items in /Library/LaunchDaemons are started at boot, while those in ~/Library/LaunchAgents are started at login.