I would like to know how much better either the tape or paste is over using nothing.
3M has many Thermal transfer tapes.
I am not sure which one would be good for this application.
Using "something" is good.
Choice is as much based on mess, ease and cost as effectiveness - as most systems that work at all work well enough in most cases.
Usually Rth_sa (thermal resistance heat sink to air) is the main factor.
Rjc (junction-case) is set by the manufacturer and is usually good enough if you do your part OK.
Rcs (case-sink) which you are asking about here is usually not the major factor as long as you do it half well. It's usually only in super high power applications that Rjc and Rcs matter crucially as then Rsa needs to be very low and the rest start to matter more.
As well as what you mention you can get thermally conductive rubber washers - these are what I usually use and they have the advantage of electrical isolation as/if desired.
The advantage of any jointing method designed for the purpose is that it provides intimate contact with heatsink and device and avoids air voids. Air is a poor heat conductor and quite small air voids across a surface can noticeably increase thermal resistance. Using no compound etc makes consistency harder to obtain.
Manufacturers provide data on performance but you can easily check by making several devices which are the same except for jointing material, running each at the same and highish wattage and measuring device to sink temperature drop. If you can comfortably touch the IC at full cry when temperature has stabilised it is usually OK enough - under 55 degrees C case if you can hold a thumb on it for many seconds. For first test dab with a wetted finger or other material - to avoid leaving skin on the device.
A mere whiff of forced air flow makes a vast difference to heatsink performance. In purely passive systems working on helping air flow can also help muchly. Avoid arrangements which block flow unnecessarily.
If you intend to mount the heatsinks as per photo, strongly consider not doing so. In an ideal vibration free environment they may last OK but in most real world conditions that is risking fracture of the leads with time.
Better would be a single heatsink on one side extending down to the PCB and screwed or otherwise anchored to the PCB in some way. Even attachment with eg neutral cure silicon rubber would be better than waving in the breeze. A single heatsink may also give some gains in heat transfer per area. Mount the TO220 device with short leads near the PCB so the majority of the heatsink is above it.
The long leads on the power resistor at rear helps prevent board charrin in extremis BUT makes resistor cooling worse. You can get a useful amount of cooling from PCB pads when the leads are short and copper area is maximised. For reliability resistors should be run at 1/2 max power or less so usually temperatures should not be so high as to damage PCBs.
If the regulator drops substantial voltage always consider using a series resistor to move heat out of the regulator. The resistor should be sized such that the regulator has enough voltage headroom at Vin_min and Iout_max. A 5W ceramic bodied micro-brick style resistor dissipating say 2 Watts is easily air cooled and takes 2 Watts off the cooling needed by the regulator. A 10 W resistor can happily dissipate up to say 5 W and if you need more than that your design is usually suspect with LM317 etc. LM350 at full cry is harder.