Interesting question. As a general answer, Poverty is poverty and it needs to be tackled globally and not just focused on regions. The well being of humanity should be the pinnacle point and should be tackled as a global matter.
But the poverty experienced in third world countries is a whole different ball game. Most of the countries/town/cities in these regions suffer from additional issues which come with poverty. They have things like famine, incurable diseases, genocide, war, unwanted dictatorship and high levels of inequality. How many of these are large threats in westernised countries? If you take a "beggar" in for example Zimbabwe. He/She is under the rule of quite possible the most disgusting man in modern day politics, Robert Mugabe. They are under hyper inflation where any kind of money they get is not even enough to buy a slice of bread. The widespread of Aid's/HIV and other diseases is so severe, not mentioning they don't even have any kind of regulated health care service to protect them, or even state benefits. If one of them suffer some kind of illness and collapse on the street, more than likely they will be left for dead. The worst is, children are suffering as much as adults. There are charities such as the Red Cross, Unicef, and other NGO's which help - but how much can these non-profit organisations do? When Zimbabwe was going through it's most troubled period in 2008 their government was stopping western aid to the country. How can these people live without, what is at times, there only source of survival?
Let's take an average begger in for example the UK (where I live). He/She is living on the streets and at times can make a fair bit of cash through begging. The majority of beggers you see are drunks or drug addicts, so the question needs to be asked where does this money they get given go to? They have a number of charities and Initiatives from institutions like Churches now helping them out which give them hostels to sleep in for the night and a warm dinner. If, god forbid, a beggar suffers any kind of life threatening condition and collapses on the street, no doubt an ambulance will be on it's way to help him/her. Would this happen in a third world country? I doubt it.
Have you seen some of the benefits the "poor" can also get, especially in the UK? Now that's shocking. Some people on benefits earn more than an average person in a full time job!