It is mostly about oversampling. And you need to look separately at Renoise instruments a.k.a. Renoise realtime sampler engine vs. VSTis, which provide their own engine. If you have a quite old VSTi, it may not use oversampling (e.g. play NI pro53 at very high notes, you will hear aliasing noise, which I actually like). If you then double the sample rate, the VSTi will render like it was using oversampling 2x, since the time resolution now is doubled while actual calculation, and it will be resampled later back, outside the synth, with a lot of oversampling.
If you look at Renoise instruments, it may behave different. I think dblue stated once that realtime mode render should sound exactly like realtime (not sure though, I sometimes imagine slight differences, but could be a VSTi instead), and the other offline modes also try to optimize the resampling quality of the samples, so can sound slightly different (usually a bit more bright and precise at high end). If you used drum samples played and sampled at 44,1kHz and now are switching to 48kHz output, it could even result in “worse mathematical quality”, since now the sample needs to be resampled and not just played in original state. Depending on your samples resample setting, it can the sound more or less drastically different. This scenario vice versa, like you described it, you were already used to the resampled sound of your drum samples, switching then to 48kHz may result in some more dull yet more precise and less noisy sound.
A modern VSTi usually provides oversampling and other techniques, so I don’t think there will be much audible difference when switching sample rates, as long you are not going below 44,1kHz.
If you composed your song in 44,1kHz, I would also render it in 44,1kHz, since it is meant in this way. A 48kHz usually sounds a tiny bit brighter, I guess due more precise resampling (e.g. compare Renoise cubic vs. sinc on a hihat, not played at sampled rate/output rate). I also think people are still used to the 44,1kHz sound, nothing wrong about it. I normally use 48kHz btw., mainly because also my hardware uses that as standard. So do a lot of streaming portals, too. There is less resampling involved, if you use 48kHz.
Even if you only used 44,1/16bit samples, you will highly benefit from a 24 bit render, because you usually applied fx and volume modulation and also you replayed the sample not at sampled rate, so used the resampling engine.
Some older synths and fx have a buggy implementation, so switching samplerate will also affect pitch.
If you want a 1:1 render of your song, you can use a vst recorder like melda MRecorder (free).
I couldn’t hear any quality problem in your productions btw.
Oversampling does not necessarily will sound “better”, it will sound “less noisy”. Noise can be beautiful though. Resampling obviously sounds different in Renoise. But also I am not sure if you can say it sounds “worse” than original samplerate.