We have a competitor that copies our features, and often says they can do the same thing as us, even if they cannot. How do you approach this?

This is a tale as old as time. There are many ways to approach this.

I have seen vendors heavily bid on the AdWords of their competitor names and promote alternative solutions (trials or marketing content). I have seen vendors who are challengers in mature markets create "Us vs. Them" web pages or blogs that outline the differences (according to the vendor publishing the info) what the major differences are. And, I have seen third-party research or analyst evaluations heavily promoted that rank vendors according to specific criteria (contracted by the vendor or annually conducted created by a research firm). 

However - the most powerful thing I see is customers doing the work for you.

In general, I'm not a fan of tearing down competitors publicly. Think about a time you heard a friend talking about someone behind their back - it makes you feel uneasy, maybe even distrustful of your friend, right? 

Now think of the inverse. 

What if you heard you friend tell you a positive anecdote about a service or new app they used for a problem you have. How likely are you to try that service/app? I believe positive associations are more powerful than negative associations for long term engagement and trust. 

Customer stories are your biggest ally here. Maybe you can include in the customer story that Customer A replaced an *unnamed* competitor with your solution, and UNLIKE *unnamed* competitor, they derived the following three benefits and business outcomes with your product. And now you can arm your sales team and/or partners with this story to have a specific "you vs. your competitor" conversation with one customer at a time, proven out by a trial or proof of concept.

Put your customers on your website, have them speak to other customers (video testimonials, backchannel phone calls, on stage at your conference, etc.). Customers are your biggest offense against competitor messaging. 

In this case, be competitor-aware, but customer-focused. 

You don't have much control over what your competitor’s teams are doing. However, what you can drive is focusing on delivering business results for your customers and getting your internal company teams to work with your customers to publish these stories, case studies and testimonials.

With this approach, your new customer prospects may evaluate you and your competitor and at first, might think you both offer the same features. However, you can now easily turn the conversation from features to driving outcomes. You will have your customers publicly highlighting how you helped them succeed; which is what your prospects want to achieve.

By focusing on consistently winning for your customers, you can shift the narrative from feature to business value.

While it may be tempting to get sucked into feature by feature wars and spar through marketing communications it will not yield much for whoever (you or the competito) lacks sources of Sustainable Competitive Advantage. see related answer on what those are, not to be confused with Competitive Advantages