My camera offers both a mechanical and an electronic shutter and I cannot see any difference

For a static subject in steady light they genuinely produce the same picture, which is why your test showed nothing. The differences appear in specific conditions.

Electronic shutter advantages:

• Silent. No moving parts, no sound at all. This is transformative for theatre, ceremonies, wildlife and anywhere a click is intrusive.
• No vibration. Nothing moves, so there is no shutter shock softening long-lens or macro shots.
• Very fast top speeds and high burst rates.
• No mechanical wear. Mechanical shutters have a rated life measured in actuations.

Electronic shutter problems, and these are the ones that catch people:

• Rolling shutter distortion. The sensor is read out line by line rather than all at once, so anything moving fast across the frame is skewed. Propellers bend, golf clubs curve, a panned car leans.
• Banding under artificial light. Many light sources flicker, and reading the sensor progressively records different brightness on different lines. You get horizontal stripes.
• Flash is usually unavailable or severely limited.

There is a third option on many cameras that is worth knowing about because it is often the best default: electronic first curtain.

It starts the exposure electronically and ends it with the mechanical shutter. That removes the vibration from the shutter opening — which is the part that causes shutter shock — while keeping a mechanical closing, which avoids most rolling shutter effects.

The catch is that at very fast shutter speeds with a wide aperture it can produce uneven exposure or oddly shaped out-of-focus highlights, because the electronic start and mechanical finish do not travel identically.

So a reasonable set of defaults:

• Electronic first curtain for most work.
• Full mechanical for very fast speeds with fast lenses, and whenever you use flash.
• Fully electronic for silence, and for static subjects where vibration matters.
• Avoid fully electronic for fast motion and under artificial lighting.

Worth adding that how bad the electronic shutter's problems are depends enormously on the sensor readout speed, which varies hugely between cameras.

A stacked sensor in a recent high-end body reads out fast enough that rolling shutter is nearly undetectable and banding is much reduced. An older or smaller sensor can take a substantial fraction of a second, and the distortion is dramatic.

So advice about electronic shutters is very camera-dependent, and the way to find out where yours sits is a two-minute test: photograph something moving fast across the frame — a passing car, a fan — and look for skew. Then photograph a wall under indoor lighting at a few different shutter speeds and look for banding.

Do those two tests once and you will know for the life of the camera which mode to reach for.

One practical footnote: the silent mode is silent enough that people around you do not know you are shooting, and that is worth thinking about beyond the technical question.

In some settings — ceremonies, performances — that is exactly the point and it is a courtesy. In others, being visibly and audibly a photographer is what makes people comfortable, and a camera that gives no signal it is working can read as sneaky.

Several cameras offer a synthetic shutter sound for this reason, which sounds absurd and is genuinely useful.