Just to add to the confusion there are CBR and VBR encoding options. CBR = constant bit rate, VBR = variable.
A 128kbps VBR file will tend to sound better than a 128kbps CBR file.
Reason: CBR literally uses 128kbps each and every second, whatever is going on in the recording. VBR uses a guideline bitrate overall, but is allowed to trade-off within that, for instance by allocating very low bitrate to silence or a low frequency 'mono' passage, adding the 'spare' to an adjacent high-frequency 'wide stereo' passage (you can safely ignore that there's also ABR, 'average' bit-rate, looser than CBR, stricter than VBR, - it's little used).
If encoding files for a particular purpose, check the destination accepts VBR files. A few years ago a large minority of software and hardware had problems with VBR - these days (in my experience) VBR is almost always OK to use and really does offer more bang-for-your-kilobit.
By the way, some VBR encoders offer a 'target' bitrate like 128, others only specify quality settings like 'highest','medium' etc, but you soon get used to what bitrates they tend to approximate to. LAME itself has a bunch of
handy presets which are in common usage - you'll see people just write that a file is "-v0".
I personally always use -v0 (--preset fast extreme), 245kbps VBR - it's about as good as mp3 can get. I'll not go below -v2 (190kbps VBR) for anything I intend to be listened to offline, though online players often require a lesser bitrate. Where a very low bitrate is needed I'll encode the file to mono, as loads of the lo-fi artefacts arise from corruption of the stereo image.