The differences that you are hearing are due to a whole range of things. I can't find a Stagg DB60 in the current Stagg range. Would it be similar to the current Stagg 60BA? (60w Combo, 12" speaker, B, M & T controls, compressor)
http://www.imuso.co.uk/Amplifiers/Ampli ... fier-ComboIf it is, then already you are talking about a difference in loudspeaker quality. Mr Stagg can't afford to spend too much of his budget on an expensive bass-handling loudspeaker if the whole amplifier is only going to cost about £100. Is the Vox loudspeaker in a bigger box? That will also make a difference.
The effect of the compressor in your amp will only be apparent if you play it loudly. Its function is to stop the amplifier from going into harsh distortion by 'compressing' the peak sound levels into something that the amplifier can manage. It also protects the loudspeaker, but it does reduce the dynamics that the player can use as a form of expression. However, some players like it because it enables them to play at a more consistant level, never too loud, never too soft.
As your amp is designed as a bass amp, you
should get a satisfactory sound with all the tone controls in their mid-position. Don't forget that a bass guitar is not an instrument to be played in isolation, so hearing it in a group context may make it sound different as it blends with other instruments (and you'll be playing it loudly, which is what the speaker is designed to do). Also, if the amp is new and hasn't been used at any great volume, then the speaker cone surround hasn't been flexed very much. A bit of 'running in' at good volume will help to 'loosen up' the speaker and may improve the sound.
In the end it comes down to educating your ears so that you know what you are listening to and can work out how to improve it. You may well end up with a more expensive amp/speaker but simply changing to valves or using an effects pedal is not going to make the difference that you hope for.
Ray