Selling to a profit center
If you are selling a tool to a profit center, you are enabling your customer to do more faster. To be somewhat reductive, this means that you are selling a time saving measure.
Selling to a cost center
If you are selling a tool to a cost center, then you are enabling your customer to do the same amount of work cheaper.
The mental model
Think about your customer's product as a vector in a 2-d space where the axes are "time savings" and "cost savings".
The magnitude of your customer's product vector is the important part. A larger magnitude means "more success", while a smaller magnitude means "less success", however your customer defines "success".
Your product is also a vector in that same space.
The magnitude of the sum of your customer's product vector and your product's vector is the success that your customer can have when they're using your product. This is obviously what you should try to optimize. This is shown as the dotted line in the above illustrations.
It doesn't matter which axis is increasing more because of the influence of your product's vector.
This is because the same product can be sold as a time saving tool to a profit center, or as a cost saving measure to a cost center. The trick is to figure out if your target user is a cost center or profit center and then frame your product as the appropriate type of tool. Selling tools to developers is usually selling to a profit center. This means that if you are selling to developers, the entire experience around your product must be quick and snappy.
Your product should be installable in either a single command or two, or be available to download, install, and use right away - without needing to have a back-and-forth with a salesperson.
Your documentation should have a quickstart guide that enables your users to get up and running in a matter of minutes.
The more quickly that a user can be successful with your product, the larger the time savings offered by your product will seem. A larger (perceived or actual) time savings makes selling your product to a profit center that much easier, because you are already showing evidence of your product's value proposition.
When you are building a product, you are actually building two correlated products; your actual product, and the experience around using your product. Clearly, it is vitally important that your product actually solves your customer's issue. However, your product must also solve that issue in a way that is better along either the "cost savings" or "time savings" axis than any other solution. It is your job to figure out which of those axes is more important to your customer, based on whether they are a cost center or profit center, and market your product accordingly.