There are few reasons:

• Sauron is REALLY afraid that any moment his opponents will use the One Ring and become much more powerful. He currently has the advantage but he might loose it any moment, so he attempts this blitzkrieg instead of regular siege.
• Mordor's orcs are too undisciplined for longer siege: sooner or later they would start fighting between each other (if they wouldn't have opponent to fight with), which means every day would reduce the power of his troops.
• Minas Tirith is a fortress, which means it is prepared for long siege with supplies (or even inner source) of food and water. His huge army would have serious problem with providing logistical supply
• He has huge advantage of numbers and power so he believes in quick victory.
• Taking the city quickly would give him a chance to defeat the Gondor before Rohan would have a chance to respond
• Saruman is confident in the power of his new Uruk orcs and his technology. He also has a cunning plan to break the Helm's Deep fortifications.
• Saruman probably wants to show off to Sauron, possibly to prove that he would be not a mere servant but maybe a someone with almost equal power.
• It would look VERY bad for him, if Sauron had taken Minas Tirith quickly while he was still besieging the Hornburg.
• (As pointed by @Scott) Saruman had Eoden exiled, greatly reducing the number of cavalry in Rohan's army. But exiled or not, Saruman could safely assume that Eomer would probably return (even without Gandalf's guidance) to harass the besieging orks. So quick attack would greatly reduced defender's power and Saruman's loses.

In other words, for both Sauron and Saruman have a solid reasons for quick attack instead of long siege.

There are a few reasons:

Sauron

• Sauron is REALLY afraid that at any moment his opponents will use the One Ring and become much more powerful. He currently has the advantage but he might lose it at any moment, so he attempts this blitzkrieg instead of a regular siege.
• Mordor's orcs are too undisciplined for a longer siege: sooner or later they would start fighting amongst each other (if they didn't have an opponent to fight), which means every day would reduce the power of his troops.
• Minas Tirith is a fortress, which means it is prepared for a long siege with supplies (or even inner source) of food and water. His huge army would have serious problems providing logistical supply.
• He has a huge advantage of numbers and power so he believes in a quick victory.
• Taking the city quickly would give him a chance to defeat Gondor before Rohan would have a chance to respond.

Saruman

• Saruman is confident in the power of his new Uruk orcs and his technology. He also has a cunning plan to break the Helm's Deep fortifications.
• Saruman probably wants to show off to Sauron, possibly to prove that he would not be a mere servant but maybe someone with almost equal power.
• It would look VERY bad for him, if Sauron had taken Minas Tirith quickly while he was still besieging the Hornburg.
• (As pointed out by @Scott) Saruman had Éomer exiled, greatly reducing the number of cavalry in Rohan's army. But exiled or not, Saruman could safely assume that Éomer would probably return (even without Gandalf's guidance) to harass the besieging orcs. So a quick attack would have greatly reduced the defenders' power and Saruman's losses.

In other words, both Sauron and Saruman have solid reasons for a quick attack instead of a long siege.

There are few reasons:

• Sauron is REALLY afraid that any moment his opponents will use the One Ring and become much more powerful. He currently has the advantage but he might loose it any moment, so he attempts this blitzkrieg instead of regular siege.
• Mordor's orcs are too undisciplined for longer siege: sooner or later they would start fighting between each other (if they wouldn't have opponent to fight with), which means every day would reduce the power of his troops.
• Minas Tirith is a fortress, which means it is prepared for long siege with supplies (or even inner source) of food and water. His huge army would have serious problem with providing logistical supply
• He has huge advantage of numbers and power so he believes in quick victory.
• Taking the city quickly would give him a chance to defeat the Gondor before Rohan would have a chance to respond
• Saruman is confident in the power of his new Uruk orcs and his technology. He also has a cunning plan to break the Helm's Deep fortifications.
• Saruman probably wants to show off to Sauron, possibly to prove that he would be not a mere servant but maybe a someone with almost equal power.
• It would look VERY bad for him, if Sauron had taken Minas Tirith quickly while he was still besieging the Hornburg.

In other words, for both Sauron and Saruman have a solid reasons for quick attack instead of long siege.

There are few reasons:

• Sauron is REALLY afraid that any moment his opponents will use the One Ring and become much more powerful. He currently has the advantage but he might loose it any moment, so he attempts this blitzkrieg instead of regular siege.
• Mordor's orcs are too undisciplined for longer siege: sooner or later they would start fighting between each other (if they wouldn't have opponent to fight with), which means every day would reduce the power of his troops.
• Minas Tirith is a fortress, which means it is prepared for long siege with supplies (or even inner source) of food and water. His huge army would have serious problem with providing logistical supply
• He has huge advantage of numbers and power so he believes in quick victory.
• Taking the city quickly would give him a chance to defeat the Gondor before Rohan would have a chance to respond
• Saruman is confident in the power of his new Uruk orcs and his technology. He also has a cunning plan to break the Helm's Deep fortifications.
• Saruman probably wants to show off to Sauron, possibly to prove that he would be not a mere servant but maybe a someone with almost equal power.
• It would look VERY bad for him, if Sauron had taken Minas Tirith quickly while he was still besieging the Hornburg.
• (As pointed by @Scott) Saruman had Eoden exiled, greatly reducing the number of cavalry in Rohan's army. But exiled or not, Saruman could safely assume that Eomer would probably return (even without Gandalf's guidance) to harass the besieging orks. So quick attack would greatly reduced defender's power and Saruman's loses.

In other words, for both Sauron and Saruman have a solid reasons for quick attack instead of long siege.

There are few reasons:

• Sauron is REALLY afraid that any moment his opponents will use the One Ring and become much more powerful. He currently has the advantage but he might loose it any moment, so he attempts this blitzkrieg instead of regular siege.
• Mordor's orcs are too undisciplined for longer siege: sooner or later they would start fighting between each other (if they wouldn't have opponent to fight with), which means every day would reduce the power of his troops.
• Minas Tirith is a fortress, which means it is prepared for long siege with supplies (or even inner source) of food and water. His huge army would have serious problem with providing logistical supply
• He has huge advantage of numbers and power so he believes in quick victory.
• Taking the city quickly would give him a chance to defeat the Gondor before Rohan would have a chance to respond
• Saruman is confident in the power of his new Uruk orcs and his technology. He also has a cunning plan to break the Helm's Deep fortifications.
• Saruman probably wants to show off to Sauron, possibly to prove that he would be not a mere servant but maybe a someone with almost equal power.
• It would look VERY bad for him, if Sauron had taken Minas Tirith quickly while he'd be still besieging Hornburg.

In other words, for both Sauron and Saruman have a solid reasons for quick attack instead of long siege.

There are few reasons:

• Sauron is REALLY afraid that any moment his opponents will use the One Ring and become much more powerful. He currently has the advantage but he might loose it any moment, so he attempts this blitzkrieg instead of regular siege.
• Mordor's orcs are too undisciplined for longer siege: sooner or later they would start fighting between each other (if they wouldn't have opponent to fight with), which means every day would reduce the power of his troops.
• Minas Tirith is a fortress, which means it is prepared for long siege with supplies (or even inner source) of food and water. His huge army would have serious problem with providing logistical supply
• He has huge advantage of numbers and power so he believes in quick victory.
• Taking the city quickly would give him a chance to defeat the Gondor before Rohan would have a chance to respond
• Saruman is confident in the power of his new Uruk orcs and his technology. He also has a cunning plan to break the Helm's Deep fortifications.
• Saruman probably wants to show off to Sauron, possibly to prove that he would be not a mere servant but maybe a someone with almost equal power.
• It would look VERY bad for him, if Sauron had taken Minas Tirith quickly while he was still besieging the Hornburg.

In other words, for both Sauron and Saruman have a solid reasons for quick attack instead of long siege.