There are definitely differences, but they are mostly pretty subtle, and not proportionate to the price difference.
IME and IMO, the main reasons for buying an expensive bass have to more do with esthetics and pride of ownership. There are lots of examples on record of great music made with inexpensive instruments. People like JJ Cale, Paul McCartney and dozens of others immediately come to mind.
That said, tuning is faster and easier on a high end bass with Hipshots or Gotoh GB7's, as opposed to a cheap Squier with elephant ear tuners. Tuning stability may also be better. Gently resting my hand on the neck of my $299 Squier moves the tuner needle slightly, whereas I have to push quite hard on the stouter necks of my good basses to get the same effect.
Shielding may be better, although many offshore instruments are nice and quiet these days. The Neutrik and Switchcraft jacks and more expensive pots on good basses tend to last a little longer, and feel more solid.
IME, the feature sets on high-end basses tend to bias them toward a more articulate, modern tone. The mid-priced Squier, Epi and Ibanez project basses I've had the last few years tended to sound a little darker on average, compared to the boutique basses I've owned or sampled.
However, it's pretty easier to move your tone in one direction or the other with string or electronics changes these days, so that's not a show stopper. Even if you add pro touches like straplocks or heavy-duty pots to a cheap bass, the cost savings are still substantial.
As others have said, rock what you got. More time spent practicing and playing out = less time for mods, and better results in the long term...