The 5-speed in the WRX is sufficient if you don't drive too aggressively, stick with a stock clutch, and keep power levels somewhat near stock. For those situations where these requirements are not held, it will eventually break. The quality of the driver and the care taken can change that eventuality to within a few hundred miles to never and anywhere in between. Still, the WRX's 5-speed is certainly the main hurdle to surpass when trying to upgrade the car to higher levels of performance.
The 6-speed STi transmission and it's related differentials are extremely robust parts for a street driven 4-door compact car. The STi often sees motorsport use in situations where non-streetable power is not needed with very little modification. Ice racing, rally racing, autocross, all manor of amateur motorsports, it's an extremely well suited car from the factory in no small part due to the strength and flexibility (six short gears) of the drivetrain.
As a possible upgrade for the WRX, it does make some sense. However, the rest of the WRX drivetrain cannot simply be hooked up to the STi 6-speed. Also, the STi 6-speed itself is quite pricy costing as much as $6,000 for a good one, that part alone. Because of the high demand for modification on the WRX and the high rate of WRX 5-speed transmission failures, there are a variety of options available to WRX owners outside of swapping the entire driveline over to the STi's setup. In fact, the STi 6-speed swap into a WRX will likely exceed the cost of selling the WRX and buying the STi with that setup already installed. As such, most people will recommend "If you want the STi's transmission, buy an STi". There are other upgrades between the WRX and the STi (turbo, brakes, DCCD, etc) but the transmission is the most costly.
The aftermarket 5-speed options then are pretty appealing choices to most WRX drivers. They come in a variety of shapes and sizes, the best of which can exceed the STi's 6-speed in strength and shock loading capability. Also, a custom transmission lets the driver choose a gearing that makes sense for their chosen areas of interest (IE, if you want to focus on drag racing). There are also somewhat budget oriented options aiming to get the typical ~300whp WRX back on the road cheaply and reliably.
Do you need to tune your engine for a transmission change? No. Unrelated. Though you may not be considering all the computer related fun of trying to get DCCD to work in a WRX after swapping in the STi's 6-speed. It's a fun process.
Acceleration depends on several variables. You generally want to shift as little as possible between where you start and where you finish. Custom built drag cars typically have only 2 gears. One to get them off the line, the other to make it from there to the finish line. Of course, those cars will be going hundreds of miles an hour by the finish line. You would not want to build a WRX with that kind of thinking. First off, it's a terrible choice for a pure drag car (AWD, 4dr, manual, etc). Second, drag cars aren't much fun for anything other than accelerating in a straight line. If you want to use that round thing in front of you from time to time, you're going to have to consider gearing that makes more sense to start accelerating at other speeds than zero. The STi's 6-speed is considerably better suited than the WRX's 5-speed for several common speed increments. What does that mean? It means the STi's second gear stops at 55mph so it's 0-50 time is better than a WRX. 0-60 requires 2 shifts on the STi, it's not that good a choice for the 6-speed. Also, the STi is going to be faster from cruising RPM's for most of your typical 20-40mph, 40-60mph, 60-80mph, or any such type speed increments. It will have more speeds that sit nicely in the fat part of it's gearing (since it can choose between more gears over those speeds) when acceleration is not just defined as from 0 to as fast as you can go.
All that said, if you wanted a pure 1/4 mile oriented gearing on a WRX, it's going to look pretty tall, taller than anything sold by Subaru. The WRX's 5-speed is considerably taller than the STi's 6-speed. In fact, 6th on the STi is SHORTER than 5th on the WRX (4th on the WRX and 5th on the STi are pretty similar). Rally racing calls for some of the shortest gearing of any motorsport. Drag racing calls for some of the tallest gearing of any motorsport. Hopefully this helps explain that difference.
2015 STi: Still Stock
04 STi, 04 WRX: SOLD