A 4-stage RO/DI system is typically one sediment filter, one carbon block, one membrane, and one DI stage. A 6-stage system is usually one sediment filter, two carbon blocks, one membrane, and two DI stages. The bells and whistles (pressure gauges, in-line TDS meters, booster pumps) are mostly extras and may or may not be required. When properly maintained, both will produce the exact same quality of water. There are, however, certain benefits to using a 6-stage as opposed to a 4 stage.
To understand if a 6-stage would be better for you than a 4-stage, it helps to understand what the stages do.
First is the sediment filter. The sediment filter is designed to remove large particulate matter. This protects the carbon blocks from plugging up and becoming useless before they can no longer handle chemicals like chlorine/chloramine. Both systems only have one of these, so there's no need to do a comparison here.
Next is the carbon block. The carbon block removes chlorine and can break and remove chloramines. Chlorine is really hard on TFC membranes, so having good carbon blocks and changing them appropriately is essential to getting the most life out of your (relatively) expensive RO membrane. The 6-stage system likely has two carbon blocks while the 4-stage only has one. If you have two carbon blocks and the first one exhausts, the second can pick up the slack for a while. If you only have one carbon block and it exhausts, you're sending chlorine and chloramine straight through to your RO membrane, which can damage it prematurely.
The prefilters (sediment filter and carbon blocks) are really all about protecting your membrane. You will not see a measurable increase in water quality from more prefilter stages. You will, however, be able to get away with less frequent carbon block changes, and your membrane will likely be protected for longer. I run two carbon blocks and like doing so.
The third part in your system is the RO membrane. Both the 4 and 6 stage systems will likely have the exact same membrane, so no comparison can be made here. Again, just keep in mind that a 6-stage system will protect the membrane a bit better.
Last is the DI stage. If your RO membrane removes 95% of the junk in the water, the DI will remove the last 5% or so. Having one DI stage is fine and will work without issue. If you have two DI stages, however, you can swap canister 2 into canister 1's place, and put fresh resin in canister 2. This will make sure that you always use all of the resin's exchange capacity before throwing it out. Your TDS might start to rise if you have only one DI stage even before the resin is fully exhausted.
In terms of extras, I don't particularly like inline TDS meters. They're relatively expensive and relatively inaccurate when compared to a quality handheld meter that temperature compensates (the inline meters, by and large, do not temp compensate). The handheld meters require more work to use though. You'll have to decide for yourself if the inaccuracy and money spent is worth the convenience. You also don't need a booster pump unless your PSI is under 50. My PSI runs around 45 and I still don't need a booster pump to get good rejection rates (> 96%). Pressure gauges actually can be useful though. I would put one before and after your prefilters. This can tell you when it's time to replace your sediment filter (or carbon blocks, if they happen to get plugged).
Personally, I prefer to run 6 stages over 4. You will likely not get measurably better water, but it gives you a bit more breathing room on filter changes. If you'd like, you can always upgrade your system by adding two canisters, one for a carbon block and one for another DI stage. You can also add TDS meters or pressure gauges without buying a new system.