Learn French: Cette discothèque est très célèbre.

It is widely understood, but it can sound slightly formal or a bit dated to younger speakers. In everyday spoken French, people more commonly say une boîte de nuit (literally 'a night box') or simply use the phrase aller en boîte (to go clubbing).

What is the difference between using célèbre and connu here?

Célèbre means 'famous' or 'renowned', implying a high level of widespread recognition or prestige. Connu translates more closely to 'known' or 'well-known'. A popular local bar might be très connu in your neighborhood, but a world-class club in Paris or Ibiza is très célèbre.

Why doesn't the adjective célèbre have an extra '-e' at the end to agree with the feminine noun discothèque?

It actually is in the feminine form, but it's invisible! Adjectives that already end in an unaccented -e in their masculine base form (like célèbre, facile, and jeune) do not take an extra -e. They are spelled and pronounced exactly the same for both masculine and feminine singular.