ITIL consulting is help choosing, adapting and rolling out ITIL practices in a real IT support operation, rather than adopting the whole framework on paper. Most teams do not need all 34 ITIL 4 practices. They need incident, request, problem, change and knowledge management working together, measured honestly, and understood by the people doing the work. HDI consultants have implemented service management in support centers for more than 35 years and hold ITIL certifications through ITIL 4 Managing Professional.
Training gives your team the shared vocabulary. Consulting applies it to your operation. Teams that do both tend to get further, because the people learning the practices are the same people changing them. Talk to your account manager about pairing the two.