The advantage of SSDs is speed and shock resistance, the latter being particularly important in laptops that get shunted around a lot.
The downside is that they cost more, and that they tend to fail more suddenly and more completely than spinning rust, so there's less likely to be a chance to recover data from one when it starts to fail. The flash memory used in SSDs eventually wears out -- you can only write to each cell reliably a certain number of times -- but modern SSD controllers maximize the life of a drive with wear-levelling algorithms and make SSDs about as reliable as HDDs, in practice.
It's a while since I looked at power consumption figures for SSDs, but a couple of years ago I was surprised to see that SSDs used almost as much power as HDDs (comparing 2.5" SATA models) where I'd been expecting them to use significantly less.
2.5" HDDs do use noticeably less than 3.5" HDDs, especially when spinning up the disk, so if you want low power you definitely don't want to go 3.5".