The biggest differences will be in your riding position which changes your aerodynamics, and your tires. At 14mph, aerodynamics doesn't matter too much. So I don't think you'd gain much performance by getting a new bike. If your tires are knobbly, getting good slicks would probably make a noticeable difference.

My only bike is a 1995 steel-frame Marin mountain bike with slicks, flat bars, and a milk crate on the back. I learned to get aero on that thing and I can descend at 50mph+ on the hills here. That puts me in the top 1% on Strava. I've also gotten on leaderboards on popular flat stretches. So that tells me that with slicks and the right riding form, my bike is just as good as the fancy carbon things on flats and descents. The weight makes a difference on climbs, maybe 1-3mph on a 7-10% grade with my 40lb bike. That's a big deal if you want climb a vertical mile, but for a normal commute it's really negligible.

That said, it's probably much more fun and comfortable to ride a nice bike. Getting low on a bike that's not made for it is not so great. So if you have the cash, it could be a worthwhile investment. But I think the performance benefit would be mostly in you getting faster by just wanting to ride more.