What are these engine inlet devices used for?
The first photo shows the engine air intake screen. Its basically a screen to prevent FOD ingestion, as you noted. From CH-47 Theory of Operations:

Engine Inlet Screens
An engine inlet screen which maximizes foreign object damage (FOD) is installed on each engine. The reduction in engine power available with screens installed is negligible. The engine inlet screens have bypass panels.  These two panels are on the aft end of each screen. Quick release fasteners attach the panels to the screens. They are removed if the ambient temperature
is 4°C or below and there is visible moisture


I'm pretty sure it is supposed to be minimizes foreign object damage.

It is an All Weather Screen (AWS) intended primarily to prevent ingestion of large items that would cause foreign object damage to the compressor blades.

By Mr.Z-man - Own work, CC BY-SA 3.0, Link
The second photo shows the Engine Air Particle Separator(EAPS) unit, which in this case is the  Pall PUREair System

Pall PUREair system; image from Pall aerospace
The EAPS is there primarily to protect against ingestion of sand and dust, which, according to Why an Engine Air Particle Separator (EAPS)? can cause multiple problems like,

erosion of compressor blades

clogging turbine cooling passages

cause sulphidation, corrosion and overheating


As the article notes,

A CH-47 in the desert with only the AWS could achieve only 12 landings before both engines were changed. With EAPS, it did 13 landings with no change in horsepower. in another desert test CH-47 engines with AWS deteriorated to an unacceptable level in only 30 hours. With EAPS, there was no degradation.

In case of desert environments, EAPS is a must if you want your engine to survive any more than double digit hours. This particular type of EAPS belongs to the Vortex tube separators and uses a vortex generator to separate the sand and dust from air.

Contaminated air entering the ... air cleaner tube assembly is given a swirling motion induced by the vortex generator. This swirling motion causes the heavier dirt particle and water droplets to be thrown radially outward by centrifugal force toward the wall of the tube.


Principle of EAPS operation, image from Pall aerospace.
You can find about other types and their operation in A Comparative Study of Helicopter Engine Particle Separators