For a weekend in Vejle, location is more important than having a huge choice of hotel facilities. The city is compact, so staying centrally gives you the freedom to walk to restaurants, shops, the railway station and many of the places you are likely to visit.
Central Vejle is the easy choice
If this is your first visit, I would normally stay in or close to the centre. You can go out for dinner without worrying about transport and start sightseeing almost immediately in the morning.
Being near Vejle Station is particularly useful if you arrive by train or plan to make a day trip to Jelling. Do not worry if the railway area looks less picturesque on a map; everything in central Vejle is relatively close.
Consider the harbour
The waterfront is an interesting alternative if you prefer modern surroundings and like having the fjord nearby. The harbour has changed dramatically and gives you easy access to landmarks such as Fjordenhus and The Wave.
A little farther out can work too
Travellers with a car may prefer accommodation outside the very centre, especially if Vejle is being used as a base for exploring the surrounding countryside. Just remember that a cheaper room is not always better value if you end up driving and parking for every meal.
For a two-night city break, I would keep it simple: choose somewhere comfortable within walking distance of the centre. Vejle is at its best when you can leave the hotel and explore without making a transport plan first.