Well, that still doesn't show any specific computations that we could do comparisons with. But, there are some general things I could say.
analyzing data
Mathematica gives you an enormous number of built-in functions to help with analyzing data, as well as the ability so spin up your own bespoke versions easily. I would think that analyzing would involve things like visualization, data restructuring, applying statistical tests, and pattern discovery, to mention just a few. I can't imagine any advantage to doing these things in BASIC, and I would expect them to be much more error prone, tedious, and limited when done in BASIC. In addition, Mathematica's notebook interface provides a "laboratory" for doing this that I can't even adequately describe. Having said all of that, you've probably refined your own BASIC toolset over the decades, so maybe these things don't resonate with you.
long streams of numbers up to about 250,000 per file
This is right in Mathematica's wheelhouse. You'll have so many functions for processing, inspecting, manipulating, aggregating, extracting, and mapping over streams and lists.
I don't know anyone who uses it
I would think that there must be Mathematica users at any university. You also have this community. And there is also the Mathematica StackExchange (https://mathematica.stackexchange.com).
all the discussions of the software seem to be people doing work very unlike myself
I think you've just been unlucky in your exposure to Mathematica. I would imagine many people are using Mathematica to analyze streams of data.
I would offer this caution/caveat: programming in Mathematica will probably feel very uncomfortable if your previous programming experience has been limited to BASIC. If you are going to experiment with Mathematica to see if it fits your needs, you'll need to suppress the feelings of frustration that will probably arise in the early days. I think people get the maximum benefit out of Mathematica if they commit to actually learning some non-trivial programming skills. Rather than just looking for the shortest line to get what you want using Mathematica, spend a little time trying to learn to think like Mathematica. This will pay off enormously (in my opinion) in the future. So, if you're very satisfied with your BASIC workflow, and if you're not willing to spend some "overhead" actually learning Mathematica, then maybe don't bother. But if you're willing to endure some confusion and frustration for a few months in order to eventually achieve orders of magnitude greater productivity, functionality, and satisfaction, then you should definitely try Mathematica.
