There can be a subtle difference.
If I were creating something, e.g. writing some new programming code; putting some electrical or mechanical components together to achieve a particular objective; installing and setting up a new TV set for the first time [1]; then, if unsuccessful, I might say "I didn't get it to work" or "I can't/couldn't get it to work" [1]. (It had never worked, because it was new.)
If I were repairing something that had previously worked (e.g. repairing a previously working TV set [1]) and the repair was unsuccessful, then I would probably say "I didn't/can't/couldn't get it working". (It had previously worked.)
Note that these are subtle differences, and I do not mean to imply that the expressions and scenarios are necessarily always used that way around, not that it would be wrong to use them the other way around.
[1] Thanks to Janus Bahs Jacquet, whose comments contributed some examples and other input to my revised answer.