Both are fast enough for your purposes, but the FX-6300 is several years older.
Did you mean FX-6300 or A4-6300?
I'm surprised you found a computer with an FX-6300 for sale.
If it's an FX-6300, it uses an AM3+ socket (from 2011 or 2012). That has some advantages (if you want to install an "older" operating system like Windows 7) but maybe not anything you'd take advantage of. It should be fast enough for most adventure games. Most games depend more on the video card capability than the processor. There might eventually be some upcoming games that require a newer processor architecture, but I haven't seen any yet.
As for burning DVD's, we've been able to do that since the early 2000's. Make sure the computer has a DVD burner. You could use a USB DVD burner, but it's more convenient when it's built in.
The FX-6300 (and the A4-6300) don't have the so-called Ryzenfall processor flaws that HowToGeek discusses
***here***.