Is sciatica just a different name for tuberculosis of the hip?

No. No. Sciatica refers to irritation of the sciatic nerve and has nothing to do with septic arthritis of the hip.
No. TB is a bacterial infection that needs treatments with antibiotics and possible surgical debridement of the hip joint. Sciatica is generally due to either a herniated disc in the spine or possibly due to piriformis syndrome causing compression on the sciatic nerve.