##Likelihood $\neq$ Bayesian with flat prior
The likelihood function, and associated the confidence interval, are not the same (concept) as a Bayesian posterior probability constructed with a prior that specifies a uniform distribution.
###Different behavior when variable is transformed
The flat prior is ambiguous. It depends on the form of the particular statistic. For instance, if $X$ is uniform distributed (e.g. $\mathcal{U}(0,1))$ , then $X^2$ is not a uniform distributed variable. So it depends whether you define the flat prior for $X$ or some transformed variable like $X^2$.
The boundaries of probabilities will change when you transform the variable, (for likelihood functions this is not the case, ie. the boundaries remain equivalent). E.g for some parameter $a$ and a monotonic transformation $f(a)$ (e.g. logarithm) you get the same likelihood interval (the boundaries transform likewise) $$\begin{array}{ccccc} a_{\min} &<& a &<& a_{\max}\\ f(a_{\min}) &<& f(a) &<& f(a_{\max}) \end{array}$$
Example If you change the statistic, e.g. estimate the $p^2$ instead of $p$ ($p$ could be the parameter in a binomial of Bernouilli distribution), then a flat prior will result in two different 95% credible intervals but a Likelihood interval interval remains the same (in the sense that the intervals are equivalent, if $p$ is in the interval determined for $p$ then $p^2$ is in the interval determined for $p^2$, of course the boundaries need to be different).
Different concept: confidence intervals are independent from the prior
The confidence interval does not use information of a prior like the credible interval does (confidence is not a probability). Regardless of the prior distribution (uniform or not) the x%-confidence interval will contain the true parameter in x% of the cases. The credible interval will contain the true parameter only when the (uniform) prior is correctly describing the population of parameters that we may encounter, the interval may be performing higher or lower than the x%.
