¿Cuál es la diferencia entre Lo Siento y Disculpe ?Puedes dar oraciones como ejemplo.

When apologizing, there is no major difference in most situations.
Disculpe usually implies an acceptance of resposibility, though. So it is most commonly associated to an action performed by the speaker:
Disculpe (la molestia). ¿Me podría decir la hora?

In other words, lo siento is the closest in meaning to "I'm sorry", whereas "disculpe"(or its informal version "disculpa") is closer to "excuse me".
Depends on the situation if you can use one or the other. For instance, never say disculpe when giving condolences, ex except if you're interrupting another person.