These terms are often used in a slightly confused and confusing way. In an ecological context their use can be traced back to Holling (1973) who distinguished engineering resilience (how quickly a system reacts to disturbance by bouncing back to its original state) and ecological resilience (a measure of the amount of disturbance that a system can withstand before it shifts into a new, alternate stable state). If it does shift to a new state/regime, then just stopping the disturbance that caused it is no longer enough to undo the change- the same reinforcing processes that made it hard to change now act to preserve the new equilibrium. This is sometimes visualised as a ball rolling around inside a hollow or valley- the system can change to some extent (roll around inside the dip) but if it changes too much it goes over the hill to the next valley/dip and then tends to stay there. In a climate change context that might involve irreversible changes or changes that would be very hard to reverse due to positive feedbacks.

As I said, the term is often used differently in different contexts and by different authors, and can mean either engineering or ecological resilience (or something much looser). There's a nice paper by Angeler and Allen (2016) that sets out some clear definitions of a range a terms associated with resilience to try and bring some clarity to things...

Angeler, D. G., & Allen, C. R. (2016). Quantifying resilience. The Journal of Applied Ecology, 53(3), 617–624. https://doi.org/10.1111/1365-2664.12649

Holling, C. S. (1973). Resilience and Stability of Ecological Systems. Annual Review of Ecology and Systematics, 4(1), 1–23. https://doi.org/10.1146/annurev.es.04.110173.000245

With respect to climate resilience, Wikipedia refers to Folke (2006) and Nelson et al. (2007): " Climate resilience can be generally defined as the capacity for a socio-ecological system to: (1) absorb stresses and maintain function in the face of external stresses imposed upon it by climate change and (2) adapt, reorganize, and evolve into more desirable configurations that improve the sustainability of the system, leaving it better prepared for future climate change impacts." According to Chapin II et al. (2006), climate is a so called "exogenous control" in the social-ecological system (SES). Climate change affects the SES. If climate change impacts exceed a certain limit, a tipping point in the SES will be reached. If we exceed this tipping point, we leave the safe operating space. The system changes its state. This is when ball rolls over the hill. This was for instance the case in the Pleistocene-Holocene transition.