Method #1: Use a Google Chrome Extension
If you're an avid user of Google Chrome web browser, this is perhaps the best option for you. Install this extension which directly integrates tightly with the web interface of Google Drive giving you enough flexibility to manage and sync files across multiple cloud storage services.
Bi-directional data updating is supported by the extension. In simple words, if a file is updated on Google Drive, its corresponding file on Dropbox is automatically updated as well and vice versa. Through its unified Google Drive interface, one can access and browse completely synced files database which is replicated on multiple external storage services like Dropbox.
It also supports other popular cloud storage services like OneDrive and Box.
Method #2: Create a Custom or Premade IFTTT Recipe
If you've never used IFTTT service, you're going to love it to the core. It's not just handy for the solution we're looking for but can also be used to create tons of custom solutions to ease your online workflow. So, let's get started and see how it can be used for connecting both Google Drive and Dropbox accounts.
Step 1: Open this custom IFTT recipe for automating syncing across both the platforms.
Step 2: You'll be asked to authorize both Google Drive and Dropbox accounts so that IFTTT service can access them on your behalf to complete the integration and syncing process.

Step 3: Once, both the channels are connected and authorized, you can specify an optional (highly recommended) folder name which will be synced by the IFTTT recipe.

All the files added to this folder within your Dropbox account are automatically synced to the Google Drive account. There's another custom IFTTT recipe more or less similar to this one which provides more control while connecting and syncing the two accounts.
Method #3: Connect and Sync Through MultCloud Application
If you're not a fan of browser extensions or a web-based service, use this powerful and free cloud storage syncing application which works seamlessly right on your desktop.The application's free version is good enough for general users and enables easy connection and file syncing between Google Drive and Dropbox services.

Data syncing ensures you get the latest copy of the file regardless of the platform you fetch it from. Power users can also connect multiple storage services and can use the free space available on these platforms in a combined form.
Its scheduling feature is helpful for arranging automated large data transfer required at fixed intervals.
Method #4: Use Multiple Zapier Triggers & Actions for Data Syncing
Another handy option to connect Google Drive and Dropbox with data syncing option is Zapier. You can integrate both the services through it and can activate multiple zaps to sync files across both the cloud storage services.
One can also sync a single folder through this integration instead of focusing on the entire cloud storage space. It also enables you to create a custom trigger and actions sequence to complete the syncing task in the best possible way.
It's one of the best and user-friendly services to integrate and sync Google Drive and Dropbox data.
Method #5: Sync Files & Folders Through cloudHQ Service
The free plan of this extremely powerful and feature-packed cloud integration service can be used to connect and sync both Google Drive and Dropbox accounts. It's best suited for professionals and small businesses.
Professionals can also take advantage of custom apps that tightly integrate various Google services with external applications to save both time and money. I liked its web interface which is easy-to-use, accessible and works seamlessly.
Businesses using Google Apps must try this solution to connect and sync their data stored in Dropbox and Google Drive.
Method #6: Combine & Sync via SecureBeam Application
Another promising solution to connect and sync both these cloud storage services is SecureBeam. It can run on desktops, tablets and smartphones making it a versatile app that syncs your data no matter which device you're using it on.
Its free plan combines up to 18 GB combined space from both the storage services which is good enough for general users. Syncing is fully automated and is secure from intruders.
Its premium plans offer several vital features like fail-safe data backup and an agile support service.