Importance scores are computed within the unit of analysis (individual respondent, if HB, or within the group's utilities if LC). Then, those importance scores are averaged across the units of analysis to summarize the importance scores for the sample.
Let's imagine a 3-group latent class solution, but where three groups isn't enough groups to fully capture the differences among respondent preferences. For example, imagine that within a latent class group half of the respondents within that group prefer one level of a 2-level attribute and the other half of that group prefer the other level. The differences in preference within that latent class group of respondents will cancel themselves out (because aggregate logit-scaled utilities are estimated to summarize that group's preferences), making it appear that the (weighted) respondents within that group place no importance on that 2-level attribute.
If you increase the number of latent class groups, you will eventually be able to break those two groups of respondents out into separate classes, recognizing that they have clear differences and really do care about that 2-level attribute (but in opposite directions).
So, as you increase the number of latent class groups to really high dimensions, such as 20-group latent class solutions, or 30-group latent class solutions, the overall computed importance scores across the attributes will tend to more resemble the HB importances which are computed at the individual level and therefore tend to isolate the importance scores for the individuals within the sample.
A big question is whether to care to pay attention to attribute importances at all. The main goal typically is to use the part-worth utilities to make predictions about how the market will choose in competitive scenarios. A good Latent Class solution (with enough classes) often makes predictions about as well as HB. However, HB gives greater flexibility during analysis for filtering the results by different respondent groups and for imposing individual-level weighting.