The emails I'm going to refer to here are for individual bespoke email approaches - not volume automated HTML email marketing campaigns.
Emailing prospects has been around now for about 20 years and is viewed by some as a bit old school compared with using social media platforms such as LinkedIn for business approaches.
Is one method better than the other? Well both channels have their strengths that should be a part of a New Business Developer's toolkit. How effective each method is will depend on factors such as e.g. the service you are providing, your particular target market, the level and function of the contact you're trying to reach as well as you're writing and communication skills (very important).
A bespoke email or LinkedIn message might suit some while others may bristle, so you do need to understand and carefully consider your methods of introduction.
Main Advantage of Introducing via eMail
Better Personalisation & Branding - So you get more flexibility with message appearance. You can add company logos, images, formatting e.g. font, bold, colour, etc to help produce a more aesthetic and visually pleasing message. I think it gives a more human touch and feel as well as a more individual professional branded look.
Main Advantage of Introducing Via LinkedIn Messages or Inmail
No Spam Filter - You can go directly into a connections LinkedIn inbox or by-pass a connection request if you purchase LinkedIn Inmail. Having mutual connections that really do know and can vouch for you may help to increase your chances of a response.
My Conclusion - A Combined Approach is a Much More Effective Strategy
There's no silver bullet so to overlook one method for another is bound to decrease your chances of engagement and therefore limit how many new business opportunities you can generate.
To optimise your efforts a strategy including both bespoke emails and LinkedIn is surely a better plan. So experiment, refine, test and adapt your approach.