[Moved to the Hardware Forum]
first thing you should check is the cpu usage, if its too high it may be responsible for the the jerky playback. depending on the media bitrate, if the hdd is a slow model it might also impact performance. being past 50% full will also lower the speed of the hdd.
i don't use mac, so i cannot tell you exactly how to check things like cpu usage while premiere is running... i think there use to be a limitation for ssd's in a mac, basically it had to be a mac ssd. there are some mac compatible ssd's available from mac aftermarket sellers like OWC macsales.com, but i think i read somewhere with newer versions of mac OS its not necessary anymore. if that's true you might look at a samsung 850 evo ssd.
Firstly, may I assume that 15GB RAM was a mistype? You do mean 16GB, yes?
To address RoninEdit's suggestion above, you may want to check your CPU and RAM usage as you render. This can be done through the Activity Monitor application in your Utilities Folder. There are tabs for CPU and RAM usages. I'd suggest sorting from highest to lowest, looking at both % CPU and CPU Time measurements. In the memory tab, check total GB of memory usage.
Also - I suggest looking at Premiere's Preferences for Memory - and ensure that you've allocated sufficient RAM to Premiere, but left some RAM for your operating system, and ensure you don't have other apps running in the background, taking up resources. Any time you don't have enough RAM to do the job, the system will cache to your hard drive, and the render will slow down a lot, due to the hard drive being a moving part.
But, I think you're right in considering that your hard drive may be the culprit. You don't mention the model of Mac you're using, but those look like MacBook Pro specs. Could be iMac specs. Either way, the hard drives that came (by default) on those models are only 5400rpm or 5900rpm, unless you bought the upgrades. Additionally, 20% of available hard drive space will feel very small on a large render.
So - yes - I believe you'd benefit from an SSD. And, I believe you'd benefit from one with far more space on it. One of the 960GB or 1TB models.
Although not officially supported for Macs with your OS, Macs do not require a special brand of SSD, but with Mavericks, one had to pay some attention to a technology named Trim. You can find out more about Trim here (How to Enable TRIM For All SSDs in OS X Mavericks | MacTrast)
However, many SSD's work beautifully on the Mac without all that concern. Crucial makes SSD's with no need to enable Trim on the Mac. You may want to contact them directly, to see what they suggest. I use an older M series model which (KNOCK WOOD) has worked perfectly with my Mac since Yosemite, and I've never needed to enable TRIM. You will also find excellent choices on MacSales.com. Both of these companies are great at answering questions and customer service.
Also, Toshiba and Samsung make great models - for them, from what I gather, we DO have to enable TRIM. For that, on Mavericks, you may want to read through that article (it uses third party software or Terminal commands), but in Mac OS10.10.4, Apple finally built it into the OS for 3rd party SSD's.
Good luck with this...