Benefits of Facebook
Facebook Fan Pages are likable by pretty much everyone. This means many times that a good number of your potential likes will not be potential clients. It's easy to have a bunch of family and friends like your company page for the sake that it's your company. Honestly, there's nothing inherently wrong with that. You never know who your family and friends could connect you to. Still, a great percentage of your likes will either be out of your service area. Others may simply "like" your page through a Facebook suggestion.
One benefit of Facebook is that anyone who has something to say can post it on your Fan Page, if you allow it. Facebook is without a doubt the most public forum outside of a blog and your own website. It's easy to promote Facebook fan pages, especially when you choose a custom URL for your business page immediately (i.e. facebook.com/yourbusinessname). If you're looking for a place to start in social media, then Facebook is a great place to start. Keep in mind, many of your likes are rarely going to interact. When you do get interaction, take full advantage of it. Let them know you appreciate their feedback and commentary, at the least.
Pros & Cons of LinkedIn
LinkedIn is a professional network that is most useful for B2B (business-to-business) professionals. Yet, it is definitely good to have a LinkedIn profile for yourself no matter what business you're in. If you have a business page on LinkedIn, it's a good idea to make sure that you can claim it. Also, if you're looking to develop business connections on LinkedIn, you can create a LinkedIn group. Inviting your clients, colleagues, and business partners to this group is a good idea. It's a great way to share relevant industry info and keep in touch.
LinkedIn business pages are good for sharing relevant news items and company updates. They also allow you to ask your followers questions. Yet, not everyone has access to them. They're more restrictive than Facebook fan pages. So unless you know you have a target audience on LinkedIn, it's best to perhaps focus on Facebook. It's a great idea for you and your colleagues to interact with what clients you do have on there, however. You can get ideas from them about what sort of content they're looking for. Facebook is better for public interaction. LinkedIn is much more private, focusing on specific niches.
If you're into B2B, you'll definitely want to focus on LinkedIn while poking around on Facebook. If you're more B2C (business-to-consumer) oriented, you'll want to focus on Facebook. Either way, see what type of presence you can create on LinkedIn, as well. It can't hurt.