While there is a lot of overlap between skill sets for reps focusing on both NEW and GROWTH customers, both segment require their own unique skill sets.

Building NEW relationships is not an easy task. It takes hard work, earned trust, and a unique value proposition specific to that customer. This is ever more challenging when efforts are outbound, so the rep must have confidence but also have the ability to handle/face rejection. People may refer to these folks as 'hunters'. You've might have heard of 'smiling and dialing'.
However, with new relationship opens up new set of opportunities and use cases. If it is a new go-to-market segment in an unproven territory (e.g. Boss says: "prove New Channel Sales through XYZ Channel in XYZ timeframe"), this person needs to test the value prop, have as many interactions as possible, and learn fast. Overall, it's extremely important to understand the problem you are solving for that specific customer, and how your product can be a 'pain killer' (not a 'vitamin') for their business.

EXISTING/GROWTH business is a different beast. Becoming a 'encyclopedia' of their organization and having account plans is an important skill set for growth accounts or the so called, 'strategic account'. This requires more planning and coordination than hunting new accounts. Also, demonstrating the ability to innovate the relationship and introduce new ideas can inspire new growth opportunities. On a tactical level, it is important that you have strong customer success/delivery, and make sure the account lead has an ear to the day-to-day activities and have a presence during the culminating events. Make the primary client stakeholder look like a hero (e.g. get them promoted). They'll do a lot for you. Getting face time with strategic stakeholders is key, too.

In both cases, a common approach we've seen work for our B2B clients is by inviting the customer to co-learn on the problem/opportunity. It's the idea of 'curating proactive insights' that inspire, win trust, and help the team win new and expand existing relationships. It's an evolution to solution selling.

Not surprisingly, with both, strong sales fundaments hold true: Empathy, Process, Expectation Setting, Closing, and everything in between.

Good sales people are some of the best small business owners I have ever met. They will study their compensation plan in minute detail to determine how to make the most money in their market. No matter what you tell them to do, they will do what puts the most food in their children's mouths - simple as that. Focus your salespeople by focusing your compensation plan.

A wise man once told me there are only 3 ways to grow your business: add new customers, increase the frequency of reorders and increase your price. If you can find ways to combine these methods, the results are multiplicative, not additive.

There has been discussions that sales people are either "hunters" or "farmers". This fits into your initial description of your problem. I am not sure I believe that it is this black and white. I go back to the incentive plan. If you are not getting what you want from your sales personnel, review your sales compensation plan and run some optimization scenarios. If it isn't what you want, change the plan and then measure the effect.