I bought a small Mandevilla and transplanted to a larger pot with trellis. The Mandevilla has grown full and bloomed but isn't traveling up the trellis. Is this a different type of Mandevilla?
There isn't a "vertical climbing" Mandevilla and then a "spreading" Mandevilla -- it's all the same plant. The difference between yours and other Mandevilla vines that climb much better is probably something very simple.
If you've pruned (or "pinched") your Mandevilla this year, that would definitely result in a bushier, fuller vine that is putting its energy into spreading out rather than climbing up.
Mandevillas benefit from a high-phosporous fertilizer, so if you haven't fertilized your vine this year, that could help it start climbing.
These articles may help you:
Older varieties of Mandevilla were generally climbers. Newer varieties are being produced that have a mounding growth pattern and do not need a trellis or support.