This is the first question to answer when choosing a gym, and the one that actually decides whether a first visit works out. Both are sport climbing, but they differ in barrier to entry, in the kind of effort involved, and in whether you can do them alone.
Bouldering: none. You walk in, get shoes, and climb. There is no equipment you must know how to operate and no task you could get wrong with consequences for somebody else.
Rope: either a trained partner or an auto-belay. To climb in a pair, someone has to know how to belay, and that is taught on a course ending in a skills check. Auto-belays remove that barrier, but a minority of venues have them and they usually cover only part of the wall.
Bouldering tops out at around four and a half metres, and you land on a mat. The consequence is short, intense effort: a single problem is a handful of moves at the limit of your strength, after which you sit down and rest for a few minutes. It is closer to sprinting than to running.
A rope wall usually offers routes of ten to several dozen metres, so one ascent is minutes of continuous effort. The fatigue is different — more endurance, less explosive. There is also an element bouldering does not have at this scale: getting used to exposure, which is to say to simply being high up.
This usually settles it. Turning up at a bouldering gym alone is entirely normal and causes no difficulty — there are always people at the wall, and problems are climbed independently of everyone else. At a rope wall, without a partner and without an auto-belay, you simply cannot climb.
If you have someone to go with and you both want to learn from scratch, a belay course for two opens up far more venues than bouldering gyms alone — and rope climbing is the skill that transfers to real rock later.
Both are, in different forms. Indoor bouldering transfers to outdoor bouldering, where instead of a fixed mat you carry a crash pad — some gyms rent them. Indoor rope climbing transfers to bolted routes on rock, but needs one more step: learning to lead, meaning clipping into protection as you climb rather than hanging on a rope fixed at the top.
There is no better choice here. There is an easier one to start with, and that is almost always bouldering.
On the map, the climbing-type filter splits venues into three groups: bouldering only, rope only, and both under one roof. That third group is the most convenient starting point, because it lets you try both without changing address or buying a second pass.
Larger cities usually have several venues combining bouldering and rope. In smaller towns the only gym around is often bouldering-only — in which case the choice is made for you.