For example, your app icon would be blue and black and your logo on printed shirts would be red and black.
This is a great question and you have received a lot of advice from really wise people here. :)
I'll come straight to answering your question according to my perspective.
A brand includes a large package of things like your services, your products, your customer relations and all that you do.
A logo is basically created to represent all that in a single icon. It pictorially reminds your customers what you stand about.
When your brand expands and grows into multiple verticals, a single logo sometimes becomes less effective to convey all those messages strongly.
To create a different colored logo, or even similar looking logo with added traits, is a good approach and should be used without fear when you want to target customers that know your overall brand but are interacting with different verticals and services.
To understand more, you may want to read: http://dalepartridge.com/crazy-science-brain-sees-logo/
All the best. If you happen to have more questions, please feel free to drop a message. :)