Hi Gina ... considering old against new is not a case of comparing just 2 areas, the Levante and Poniente, as your topic title suggests. There are 3 separate areas to consider - Old Town, Levante and Poniente.
Benidorm Old Town is sandwiched between the Poniente and Levante beaches and districts, and has many shops, bars, restaurants, hostals and hotels hidden away in the little narrow streets. For more authentic Spanish culture, Benidorm Old Town is the place to go. The Old Town also has its own small beach in the cove at Playa del Malpas.
The Levante stretches a couple of kilometres east from the Old Town towards the Rincon de Loix, incorporating the area known as the New Town, which contains much of Benidorm's nightlife, and some of the larger high rise hotel and apartment complexes.
The Poniente stretches a couple of kilometres west from the Old Town towards La Cala Finestrat. Poniente is a massive area like the Levante, but nowhere near as densely populated in terms of accommodation, bars, restaurants or holidaymakers.
David