Bootstrap-$t$ still relies on assumptions for parametric distributions: If the boostrap distribution of a statistic has a normal distribution, you can use the bootstrap-$t$ method. This will lead to a symmetric CI.
If, however, the sampling distribution is skewed or biased, it is better to use the percentile bootstrap (which allows for asymmetric CIs).
Now, which method should you use?
Concerning the bootstrapped mean:
According to simulations by Wilcox (2010), the percentile bootstrap should not be used for untrimmed means (in this case bootstrap-$t$ works better); starting from 20% trimming, percentile bootstrap outperforms the bootstrap-$t$ (the situation is unclear for 10% trimming).
Another hint comes from Hesterberg et al. (2005, p. 14-35):
The conditions for safe use of bootstrap t and bootstrap percentile inter- vals are a bit vague. We recommend that you check whether these intervals are reasonable by comparing them with each other. If the bias of the bootstrap distribution is small and the distribution is close to normal, the bootstrap t and percentile confidence intervals will agree closely. Percentile intervals, un- like t intervals, do not ignore skewness. Percentile intervals are therefore usu- ally more accurate, as long as the bias is small. Because we will soon meet much more accurate bootstrap intervals, our recommendation is that when bootstrap t and bootstrap percentile intervals do not agree closely, neither type of interval should be used.
--> in case of disagreement better use the BCa-corrected bootstrap CI!
Hesterberg, T., Monaghan, S., Moore, D., Clipson, A., & Epstein, R. (2005). Bootstrap methods and permutation tests. Introduction to the Practice of Statistics, 14.1–14.70.
Wilcox, R. R. (2010). Fundamentals of modern statistical methods: Substantially improving power and accuracy. Springer Verlag.