Let's say you have a twice differenced time series. The AIC for two potential models are 2000 and 2000.1. As a rule of thumb you usually choose the one with the lower AIC, but if the model with AIC 2000.1 is a slightly simpler model would you be more inclined to choose it since both AIC's are within 2 units of each other?
Under some assumptions, AIC is an optimal model selection criterion. It strikes an optimal balance between goodness of fit and model complexity. Under these assumptions, it does not make sense to choose a model with a higher AIC value, even if the difference between the AIC values of the models under consideration is (very) small.
However, when the difference is small, you would also expect a small difference in performance. If you have to choose a single model, you would still go with the one that minimizes the AIC. But if you have an option to choose more than one model and do some sort of model averaging, a small difference between the best models would suggest both of them deserve to be part of the model combination. (It is commonly observed in forecasting that model combinations do well relative to the individual models.)