If you live in California short tubes is your only option (OEM fit and CARB compliance). Header choice depends in part on what other engine modifications you're planning or have already installed. 2013 Charger RT has the 5.7L Hemi with a 2.5" exhaust. Unless you're planning extensive power adders (e.g. forced induction, stroker kit, etc...) headers with 2.5" collector and 1 3/4" primary tubes is a good balance. Go too big (e.g. 1 7/8" primaries and 3" collector) on a stock engine and you can lose HP/TQ rather than gain HP/TQ.
Long tubes cost more to purchase and install. Typically hang lower and may cause ground clearance issues if the car has been lowered. Long tubes will benefit performance for a modified engine where increased exhaust scavenging is needed. Because of increased scavenging long tubes need a custom tune in order to keep the engine from running too lean. Gen-3 Hemi engines use Speed Density Tables to manage AFR (air-fuel ratio) and these tables need to be modified if the operating parameters are going to be beyond the factory tune.
Short tubes do not require a custom tune but my car performed smoother with a custum tune update after installing short tubes.
Cost to install shorties or long tubes is going to be exactly the same, with the exception of mid pipes. Obviously if you buy mids (catted or not) or have to have mids made, THAT is gonna cost a little more, but it's the same work to do either.
As for the difference between the shorties and long tubes, the biggest factor is going to be your end goal. How much power do you want and how many mods are you planning?
A used set of 6.4 manifolds/midpipes (plus a reducer for the midpipes to catback) will do most folks just fine. There's a decent gain to be had there too.
As for actual "shorty" headers...long tubes make more power, period. Other than living in California where the inspections/requirements call for carb numbers, I can't see how anyone could justify installing them. Again, it's the same work and almost the same price (depending on where you get them).
I'm a big fan of build once, build right.
If you have the motortrend app, go watch Episode 15 of Engine Masters where they test manifold v shorty v long tubes.
200-300$ for the headers 6.4/6.1L, but you still need the downpipes, as the headers do NOT line up with the 5.7 downpipes... Catted or non catted are around another 200-300 used in oem form, or around 150-400 for catted.
Given i was looking to pay 500-700$ i simply went with aftermarket short tubes, and i can keep the catted oem ones i have for now.