I've struggled to define the difference between the two myself and I'll try to explain what I've learnt.
Beta NFs use Fe which gives them an access to emotional atmosphere of the surroundings, not only one is aware of how others around them feel, beta NFs can easily transform these feelings. Since Fe is mostly concerned with the positive aspects of feelings, they will try to eliminate negativism and gloom and change it to more light and pleasant atmosphere. Fe in combination with valued Se adds a special intensity to one's feelings and therefor expresion of these types can seem dramatic and theatrical, their feelings are very vibrant and present in a concrete enviroment. They value people who can express themselves openly and visibly, they like to laugh loud, talk loud and enjoy passion in their lives.
Delta NFs on the other hand value Fi, feelings of these types are also very intense, but are rather kept inside or shared with selected individuals. Even though delta NFs are also very aware of the emotional atmosphere around them, they chose to rather concentrate on their own feelings or intimate relationship than on the feelings of the whole group. Deltas also value Si which adds an extra tenderness to their emotional life and forces them to seek harmony and peace rather than drama and fire.
Betas value Ni, that means evaluation of situation throughout time. They can easily spot trends and sequences that repeat themselves and calculate the most possible outcomes or future happenings. Ni works as a calculator of events. That makes them an excellent troubleshooters who can eliminate issues before they even happen. Ni however is a highly subjective introverted function and because it is a percieving function it gives them a special "outwordly" vibe. They do not see situations that happen around them, but live in their percieved visions of future as their brain automatically calculates effects of the present/past at the future. This is the reason why many poeople can see them as visionares and seers. Beta NFs however miss Te, wihich in socionics means valuation of productivity, work and quality of products. The groundings force of gamma quadra (Ni+Te) is missing in beta which makes beta NFs especially idealistic about the visions of their introverted preception. They can get easily lost in their dreams and ideas forgeting about actual activity in physical world. This is the reason why they are often called the most idealistic types of all. Along with low but valued Se they might waist time with meaningless activities and spontanues outbursts of energy followed by times of escapism and high introspection.Betas also value Ti. Ti works best in settled and organized system, they dislike chaos and anarchy and want to find structure and order where it misses.
Delta NFs value Ne, in socionis that means an evaluation of all possibilities that might or might not take a place in a physical reality. Ne understands the Ni's calculation of events, but sees it as limited and onesided. Ne hates to be constrained with just one outcome, it seeks many different options. Stratiyevskaya (one of the socionics psychologist) described delta quadra as a complex of clipped wings, which perfectly describes delta's dislike for limitations. Extroverted intuition is not concerned with seeking one concrete answer, it loves to ask questions and blooms when it can analyze the new, unusual and orginal without limiting itself with one idea or opinion.
Deltas are freespirited and open-minded people, but they value hard work and progress. They hate to waste their time feeling unproductive and lazy, they want concrete results in a concrete world, which means they are loyal and idealistic employees. Delta NFs also have low and unvalued Ti which means they can struggles with strict rules and organization. They feel at their best when they can work in open and creative environment.