How do you choose whether or not to engage with a start-up or younger companies relative to their more mature peers?

I evaluate my needs with the services provided.  Startups want to please by satisfying your requirements and not theirs.  Some established entities want to do things their way, but perhaps there's a good reason for that.  I've worked with both successfully.  No reason not to give companies a chance.  Your evaluation and experience should select the best opportunity regardless of who the service provider is.  

It might be helpful to think of yourself as a VC from two perspectives. If a VC was to buy your company what are its strengths and weaknesses and how would they grow the business in the current and adjacent market segments?If you did that on your application stack where are your holes or modernization efforts, what do your competitors have that you do not? With that in hand, a VC would go out an look at interesting companies to fill those gaps and engage them to understand, to help shape or potentially purchase now or in the future. Think of yourself as a VC and spend 5-10% of your time on those efforts. My experience is that we easily spend that time with existing partners and wonder if we're really getting the return.

First I'm looking for at what they offer and can it solve my problem. If so then I look at the company as a whole. You want to look at their credit rating, can they provide you with sales and revenue data? Do they have reference customers? Lastly what is the risk to your business if you use them and they go out of business? 

I've engaged with a fair number of startups and found it very beneficial when we were trying to solve for something that no one else had done before. Working with an early-stage company you typically have a fair amount of influence on the prioritization of their roadmap and how that fits in with the desired solution and outcomes. This agility goes a long way for future sustainability instead of getting locked into an incumbent's solution, licensing model, etc...

I don't look at the age of the business, I look at the features, services, cost, value add, and attitude of the employees behind the name.  I look at my needs and who would be able to support the successful completion of those objectives. I've worked with a start-up and will well established companies, the bottom line is have their lost their ability to build a relationship or are they just in it for the revenue?