As per my knowledge, both these sounds are considered the same in Mandarin. Do Chinese speakers them have difficulty in distinguishing between say Paul and Ball? How do they deal with it?
In the International Phonetic Alphabet, for stop consonants, the three following categories are distinguished:
(Truly) voiced stops. The vocalization begins before the release of the stop. Examples would be [b], [d], and [g]
Voiceless, unaspirated stops. The vocalization begins simultaneously with the release of the stop. Examples: [p], [t], [k].
Voiceless, aspirated stops. The vocalization begins shortly after the release of the stop. This is often perceived as a "puff of air". Examples: [pʰ], [tʰ], [kʰ].
The true situation is a little more complicated, but for any given language, we really only need to distinguish 2 or 3 of these in order to explain the phonological structure. To give some examples:
In English, our voiced stops (e.g., spelled "b", "d", and "g") are pronounced as voiced consonants; that is, [b], [d], and [g]. Our unvoiced stops are pronounced as either aspirated or unaspirated depending on context: "pop" = [pʰap], "stop" = [stap], and so forth.
In Mandarin, the distinction is between aspirated and unaspirated. There are no voiced stops at all: 白 = [pai], 拍 = [pʰai].
In many languages (including, say French), the distinction is between voiced and unvoiced, and there is no aspiration. So, you have "tout" = [tu], "deux" = [dœ].
The mismatch between different 2-way distinctions can either be unnoticable or a huge communication barrier. My experience is that English speakers have no problem communicating with either the aspiration-distinguishers or the voicing-distinguishers. If I say "to"/"do" (= [tʰu]/[du]), an aspiration distinguisher will perceive it as [tʰu]/[tu], and a voicing distinguisher will perceive it as [tu]/[du] (and vice versa).
The real problem is when aspiration-distinguishers attempt to communicate with voicing-distinguishers. This can be hard. A Chinese friend of mine complained about an Indian person who asked for "the [t]ime", and she had no idea what the "dime" he was asking about! Similarly, I've heard Europeans complain that Pinyin "makes no sense" because pinyin 'p' and 'b' "sound the same".
To address the OP's question, Mandarin speakers do distinguish a "p" and "b", but they do so on the basis of aspiration. This works fine for communicating with native English speakers, but works poorly for communicating with other non-native speakers who only make voicing distinctions (e.g., many Europeans).
As others explained, there are two qualities of these sounds which distinguish them in different languages: voicing and aspiration. I'll leave you to look up on Wikipedia what these mean exactly.
In Mandarin Chinese, b is always unvoiced and unaspirated. p is always unvoiced and aspirated.
In English, b is always voiced an unaspirated. p is always unvoiced, however it can be either aspirated or unaspirated depending on the phonetical context. It is aspirated (i.e. identical to the Chinese p) when standing alone, as in Paul, while is is unaspirated when next to another consonant as in spit.
Since p is usually aspirated in English when at the beginning of words, Mandarin Chinese speakers usually have no difficulty distinguishing it from b based on aspiration alone. Paul and ball do differ in aspiration and are easily distinguished by Chinese speakers.
It's more interesting to look at languages where aspiration can't be used at all to distinguish these two. In Hungarian (my mother tongue), p and b differ in voicing only, and they are both unaspirated. I know from first hand experience that Chinese speakers do indeed have a little difficulty distinguishing the Hungarian p and b, as I did teach some Hungarian to Chinese people I know. To them the Hungarian p and b tend to sound similar, and more like the Chinese b, because both lack aspiration. Not to say they can't distinguish them after some practice, but this was indeed a point of difficulty.
Naturally, when people learn a new language, they still keep focusing on sound differences that are important in their mother tongue. While in English p and b are (usually) differentiated by both aspiration and voicing, native Chinese speakers focus more on aspiration, and if you listen carefully, you'll notice that they occasionally get the voicing wrong in English (crab vs crap? :-) Similarly, native Hungarian speakers tend to focus on the voicing more, and will often get the aspiration wrong (they'll say Paul with an unaspirated unvoiced p). It also goes the other way: When native English (or Chinese!) speakers learn Hungarian, they'll typically aspirate initial p's, which is one of the things which makes their accent sound "English".