In my experience the only times you will really notice thin kerf vs. full kerf are:
1. When using a relatively underpowered saw. I have seen zero difference in performance with using thin vs. full kerf blades on my 3 hp cabinet saw even when doing something ugly like ripping 2 1/2 or 3" thick rock-hard old, dry oak. However I also have a late-1980s ~1 hp Craftsman 10" iron CMS that will certainly let you know you tried to cut through something larger than about a 1x4 with a full-kerf blade.
2. You will run into issues using a thin-kerf blade if you have a tablesaw with a splitter or riving knife designed for full-kerf blades. You will bind up and the results are not that enjoyable.
3. Thin kerf blades tend to be less expensive than full-kerf blades. For example, Freud's Diablo line are all thin kerf; you have to pay about 50% more to get a regular-line Freud blade that is full kerf. Ditto with pretty well everything you pick up at a BORG being thin-kerf only. The DeWalt, Irwin, etc. blades there are invariably thin-kerf only.