Lyme disease can be cured with antibiotics. The disease can, however, do damage to your nervous system that doesn't automatically go away cause the thing that caused the damage goes away.
In some people, it can trigger auto-immune responses, which similarly don't go away just cause the bacteria that caused it is gone.
This is complicated by the fact that there's a whole fake medicine ecosystem built on preying on those people and telling them that they have "chronic lyme" and that they need to undergo a whole range of unscientific "treatments" like continuous antibiotics usage, or random woo. This is enhance by the fact that there are MD/DOs willing to write scripts for these treatments as a cash cow, and the names of these Dr.s are shared online by the large community of fake science people.
The proper name for the lingering aftereffects is "post-treatment lyme disease syndrome". It seems like a pedantic difference but it's important.
It's like if a raccoon got into your house and tore everything up. If you get rid of the raccoon, you no longer have a "raccoon problem", and setting a whole bunch of traps won't clean your house up.