A question, some time ago, from my favourite Egyptologist on Twitter:
— Liz Hind (@drbhind) April 14, 2014
There's a very simple answer to this question: it doesn't just depend on who you're asking, it may also depend on what mood they're in at the time.
Of course, there are extremists on both sides of the spectrum: from the absolute purists outraged by the very idea of someone finding a use for their work, to the mathematicians who desperately claim they're not engineers in the same way that Grant Shapps claims his business was perfectly legal1, who can't see any sense in doing maths unless it leads to something tangible and profitable.
Personally, I had the kind of revelation so significant that I remember when and where I was, in late March 1999, the Monday after the clocks changed, at 8:05am in a lecture theatre in Avignon, France. Monsieur Brada, the greatest lecturer who ever lived, was explaining some of the ideas behind homeomorphisms (or something similar), in a very heavy Provencal accent, and I realised: I am an applied mathematician.
Pure maths was neat -- very pretty, it all fits together -- but I liked getting my hands dirty. If maths at St Andrews was "how fast is the train going?", maths in Avignon was "prove that there exists a vestibule". And, despite having eaten a croissant magnificent enough to make any day good, I just wasn't up for that any more.
However, my attitude has softened significantly since then. These days, it's seeing how everything fits together that makes it interesting for me these days -- playing with puzzles, finding different approaches to the same problem, generally fiddling about under the bonnet, so to speak -- that's what makes it interesting for me, most days.
Other days, it's using that knowledge to improve my mental arithmetic, or write code to solve a real-world problem, or decide how far I need to run.
The thing is, having an interest in applied maths makes pure maths more interesting to me -- and vice-versa. It's not possible for me to say one is more satisfying than the other -- they're so closely intertwined, it's like asking whether I prefer the tune or the words to my favourite song - when it's the combination.
I'll leave the last word on this to Feynman, who was speaking about Physics when he said "It's like sex -- sure, it may give some practical results... but that's not why we do it."