Main Difference

The main difference between Beggar and Begger is that the Beggar is a One who begs; one who asks or entreats earnestly, or with humility; a petitioner and Begger is a obsolete form of beggar.

Beggar

Begging (also panhandling or mendicancy) is the practice of imploring others to grant a favor, often a gift of money, with little or no expectation of reciprocation. A person doing such is called a beggar, panhandler, or mendicant. Beggars may be found in public places such as transport routes, urban parks, and near busy markets. Besides money, they may also ask for food, drink, cigarettes or other small items.
Begger

Begging (also panhandling or mendicancy) is the practice of imploring others to grant a favor, often a gift of money, with little or no expectation of reciprocation. A person doing such is called a beggar, panhandler, or mendicant. Beggars may be found in public places such as transport routes, urban parks, and near busy markets. Besides money, they may also ask for food, drink, cigarettes or other small items.

Wikipedia

Beggar (noun)
A person who begs.
Beggar (noun)
A person suffering from extreme poverty.
Beggar (noun)
A mean or wretched person; a scoundrel.
"What does that silly beggar think he's doing?"
Beggar (verb)
To make a beggar of someone; impoverish.
Beggar (verb)
To exhaust the resources of; to outdo.

Begger (noun)
obsolete form of beggar

Wiktionary

Beggar (noun)
a person, typically a homeless one, who lives by asking for money or food.
Beggar (noun)
a person of a specified type, especially one to be envied or pitied
"poor little beggars"
Beggar (verb)
reduce (someone) to poverty
"why should I beggar myself for you?"

Oxford Dictionary

Beggar (noun)
One who begs; one who asks or entreats earnestly, or with humility; a petitioner.
Beggar (noun)
One who makes it his business to ask alms.
Beggar (noun)
One who is dependent upon others for support; - a contemptuous or sarcastic use.
Beggar (noun)
One who assumes in argument what he does not prove.
Beggar
To reduce to beggary; to impoverish; as, he had beggared himself.
Beggar
To cause to seem very poor and inadequate.