In order for a tub to run on 15 amps it has to be limited to 1000 watts of heating power and a 1 HP pump motor or less. Some of our smallest tubs can plug into a 15 amp outlet, and by virtue of their small size they work well. Yes, the limitations you mentioned are true: longer heat-up after water changes, may not stay hot as long with the cover off in very cold climates, no heat when the jets are on (or at least on 'high'), and so forth. I don't know about the bubbler jets since only Caldera tubs have blowers out of all the tubs made by Watkins MFG, and they don't offer a tub which runs on 15a.
Do what Dan suggested - and I add this twist:
To the outlet in question, plug in a blow drier and turn it on high. See if that alone trips the breaker. If not - Go to the panel and begin switching off breakers (good idea to shut down your computer first) and wait until the thing goes silent. Note which breaker it is on, (turn all the other ones back on so only that breaker is off). It really should be a 20a breaker - yes, this may sound confusing, but long term a tub should run on a 20a
breaker (it will have larger wire in the walls) even though the outlet may be a 15a rated device.
15a outlet
Now go around and try all the other breakers in your place - start with the bathrooms first! Many houses are wired with one GFI outlet unit in one bathroom which is then daisy-chained to other bathrooms and the outdoor outlet. It saves the builder money, but it is a big surprise when the tub is on the patio and somebody trips the breaker by turning on a blow drier in the far bathroom.
Having said all that - if you find the patio outlet is on a 15a breaker (especially if it shares that breaker with overhead lighting) you may have some problems. Don't give up though - I have had many customers find an outlet in a bedroom or dining room near the patio which
is on a 20a circuit. If you are real lucky, that could be tapped for your tub. An electrician can put a small hole through the back of the box inside, a small hole in the wall outside, pass wire through and mount an outlet on the patio which will handle the load. This is far less expensive than you might think, and most landlords will not mind even if you move away and leave a regular (properly installed) outlet where there wasn't one before. It is cheaper/easier to mount it on the surface, but flush mounting it is really not that much harder at this point. If the patio is fully protected from rain, it could look just like any other outlet in the place -
But you may need to add a rain cover - our tubs come with this, but you can get one at a home center:
HTH