Hi Melissa,
I'm assuming you are designing a mobile-friendly version of your client's real estate website. If that's the case, it would certainly be preferable to go for a responsive design. The main reason for this is SEO. Responsive designs maintain the same URL, which keeps your "SEO juice" regardless of whether visitors are viewing your website on mobile phones or on computers.
Mobile versions of websites are created before the concept of responsive web design was fully established, and it is my guess that other real estate websites developed mobile sites long before responsive versions are mainstream. As such, it would be a good idea to develop a responsive design for your client now, to differentiate his website from others in the same niche.
Moreover, responsive design caters to all screen sizes within one CSS version, while mobile versions can only cater to one screen size. Given the variety of screen sizes throughout new and old smartphones/tablets, it would be unimaginable to create mobile versions to suit all of them.
That being said, there are considerations to be made as mentioned by Bede. You would need to ensure the backend of your client's site is suitable for easy & quick loading on mobile phones and tablets. Otherwise, a responsive design wouldn't help at all.