Does TechTool Pro do anything that Disk Utility doesn't?

As far as the main question (in your title), yes, TechTool Pro does a lot more than Disk Utility does.

In fact, in many ways they're not really comparable products. It's somewhat akin to comparing a butter knife to a Swiss army knife. That is, Disk Utility performs a number of disk-related tasks, whereas TechTool Pro performs all these and a lot of other types of tasks as well.

As for whether TechTool Pro will help you determine what is wrong with your Mac, the best guess anyone here is going to give you is "Maybe."

TechTool Pro is generally an excellent product (it hasn't had a 'perfect' history, but overall it's been quite positive). In fact, Apple even used to provide a cut down version of it to users who paid extra for AppleCare coverage. But only you can decide whether the cost of it is worth it for you or not, especially if you're planning on upgrading in the near future.

As an aside, in a 2010 iMac you may be better off just replacing the hard drive, especially if it still has its original drive in it.

From memory, if it's a 21" model it wouldn't have had the option of an SSD, so the money you'd spend on TTP may be much better spent on an SSD. It'll give new life to your iMac and you could then put it to other uses after upgrading to your new Mac. And when the iMac eventually dies due to other reasons, you could resurrect the SSD and use it as an external drive. Just another option for consideration.