Neither! Sorry. I simply don't want you to spend the time and money to come so far only to be in the still cold, windy Chicago Apartments in April or the not such a good tourist city of Austin. Chicago is much larger and has more things to see and do all year because of its age and size. I would suggest looking on Trip Advisor for both cities to see what numerous real people wrote about various things to see or do and about hotels, etc.
My honest answer would be to go to New Orleans if you want to listen to live music, eat great food and see a little of an interesting historical US city for a week in April. The weather should be just as warm as Austin. It could rain for the entire week in Austin or New Orleans or be fine. Austin is not as cool and interesting as outsiders seem to think. People who love Ausrtin often have nostalgia from spending time there going to college.
I'm required to go to Austin for one week at a time for work at all different times of the year. My office is in the heart of downtown where parking expensive and homeless, mentally ill and drugged folks are thick on the sidewalks, under bridges, etc. I've had great food at a few places in that central area but much better food in other parts of the city or surrounding cities that would require a car to reach for a tourist. If you don't want to drive or Uber everywhere, you will not be able to easily explore Austin and the surrounding Hill Country.
When there is no festival, Austin lives it's boring daily life as the Texas state capitol. Sure there is the LBJ presidential library, but I doubt that anyone would fly from South America to see a library or an university campus (the other big deal in this little city).
Traffic during commuter hours is horribly congested and slow because there are only a few main highways and streets that cut all the way through the city. Personally, I find it worse to commute about five miles from downtown Austin to stay somewhere that does not have a high concentration of homeless folks (who sadly look mostly like burned out drug addicts and who aggressively panhandle) than to drive triple that distance from downtown Dallas to the suburbs or a similar distance in Houston, Texas. I live and work mainly in the Dallas metro, so I'm not unfamiliar with homeless people. I bring it up about Austin because it goes to a whole other level there from other Texas cities I've lived or worked in.