ныть is closer to whine or to complain or to cry (producing tears and being very loud at the same time) (literally is to be painful and making you feel the pain continuosly and every second)
конючить - to complain or ask about something continuously in a very annoying way

It's up to you to decide but if you asked me I'd say that конючить is stronger than just ныть

e.g You would rather say ныть if a person's been whining (complaining etc) for not long, but if a person's been whining (complaining etc) for long (like all day, all weak, everytime etc) you'd rather say конючить

Yeah, and ныть doesn't include to ask in its meaning but конючить does so if someone is asking you annoyingly to do something you say конючить not ныть