The long-term effect of cutting down trees in large areas of rain forests is:

Large scale felling of rain forests leads over the long term to a sharply increased rate of soil erosion once the protective tree cover is gone.

Rainfall does not increase in such areas, since forest loss actually tends to reduce local moisture recycling.

Rainfall usually falls, not rises, so the claim of decreased rainfall points to a real effect yet not the erosion focus asked here.

Wind speeds may change locally but this is not the defining long term consequence of the clearing.

Without roots to bind the soil and canopy to break the rain, topsoil is rapidly washed away.

The exposed, eroded land also loses fertility, making forest recovery far harder.