Likelihood, as M. Chernick wrote, has a very standard statistical meaning. "Levels of confidence" suggest a number of statistical concepts, but are somewhat ambiguous. Reading the relevant sections in the linked article, it seems that "levels of confidence" are a subjective survey measurement taken from experts whereas Likelihood is a measure taken from a statistical model.
Likelihood is, at large, a random function telling you the probability of a particular choice of your model parameters given the data (which in a Frequentist setting is where the randomness comes from---we think of the likelihood as something that exists for all possible experiments and sets of observations). It's useful because it is instrumental in building Bayesian and Likelihoodist estimation procedures.
The paper seems to define Likelihood differently, more like the posterior predictive probability in a Bayesian setting.
So beware of confusing the paper's likelihood with classical statistical Likelihood and consider that the paper's likelihood may be a mathematical measure and "Levels of confidence" a surveyed value.