1600 and 2600 are both 6 core/12 thread chips and the 1600 is a still a great value. But so is the 2600, especially if you get it on one of the many sales going on right now.
The 2600 has slightly better (ever so slightly) IPC than 1600 but it can clock to 4200Mhz on capable motherboards, 4100 more likely though. The 1600 will probably top out at 4000Mhz, 3900 more likely. So you should be able to get a better clock speed but that 200Mhz isn't gonna make for a HUGE amount of better performance...maybe 4-5% or so.
The one thing you get with Ryzen2000 is better core boosting...it boosts more often, longer and more cores...so if you don't overclock you'll get better performance from that alone. How much depends entirely on workload or how the game uses threads.
All told, the 2600 is better but if you can't swing it the 1600 is still a great processor and well worth it's price. Also to consider is that getting a better board will let you put in a 3600, Zen2 chip, in a year that will bring the smiles back.