I find the best way to gain customers from competitors is to collaborate and affiliate with them.
Most services are niche enough that, although they seem competing, they aren’t really stepping on each other toes. While they may share the same service space, their products are slightly different and may even be complimentary. While they may seem to share the same market, their target client description is slightly different.
A successful scheme is to create a catalyst event in which your competitors are invited to speak, perform, or showcase their wares. Every business is responsibly for promoting and advertising this event to their clients. This offers a Win/Win for everyone. The advantage is that now everyone is gaining exposure beyond their individual circle of influence. You also include your products and services at this event. Now you have other service providers bringing in their clients to see your products as well as their products.
Creating a collaborative versus adversarial atmosphere often works better in the long run. In this type of environment, companies often can create a referral program in which you share revenue on clients you refer to each other. You can also make an agreement to off-load some business during your busy times to the other company (and visa-versa). If one or the other move into a different market, you are the first they will recommend to their clients.