On Piping & Instrumentation diagrams (P & ID) the flow element - FE (or flow meter) is the physical item on the pipe which measures the flow. It could be an orifice plate, or other device.
The Flow Indicator - FI -would be the gauge on the meter, which is mounted locally to read the flow. If the flow is to be read remotely, as in a control console, a flow transmitter - FT - would be needed to transmit the data. So you will often see the grouping - FE, FT, FI - on the P & ID.
As the name suggests, flow meter is a flow measuring device used to meter or (commonly said) measure the flow. it gives the volumetric flow rate in a piping system. it can vary from a simple orifice plate to complex turbine meter where the flow rate is measured as per the rotation of the turbine blades.
Flow indicator is a simple device to indicate "Flow" or "no flow" in the piping system. it doesn't give any quantitative measurement. At the best it can give the direction of flow in a piping system.
Seems like there may be two different applications for flow indicator. My experience is similar to Johnp's: the indicator is the device that lets someone see how much the meter is reading. Patricia Lougheed
