What are the differences between these two phrasal verbs and what are the best situations to use each?
These two are more or less equivalent. They can both be used for the situation where A and B speak to each other. "Speak to" can also be used for the situation where A talks and B listens without speaking.
You are more likely to encounter speak with in American English, which employs the verb + with construction (speak with, meet with) very much more than British English does.
I remember hearing a clip of Laura Bush speaking on the radio (I think to the people of Afghanistan) and saying how pleased she was to be "speaking with you" and to British English ears it sounded very odd. On the radio, how are you doing anything other than speaking to your audience?
I have seen "Speak to" being used when there is a monologue kind of a situation, A speaks and B listens. Refer Shinto's answer. This has reference to instructing, reprimanding and situations like this.
"Speak with" is a more neutral kind of a term implying a bidirectional communication.
hawbsl is right. You are more likely to encounter speak with in American English.
It's not really used in the UK. I have only heard speak with being used by Americans, on American TV programmes and films. I remember this because it was so unusual and odd that it surprised me.
It's the same with the words meet with. This one was particularly odd to me, because the word meet on its own is normally enough. It puzzled me for a while until I heard it used in other American TV programmes and I researched the use of it. From a British perspective, it is an unnecessary extra to add the word with to the word meet.
To answer your questions directly, it seems that the differences between these are in their use. The best situations to use them will depend on whether you are speaking American English or not. You can use them if you speak American English, but otherwise they are not necessary and sound odd — particularly to English and other British people.
This answer has two portions, one on to speak with/to, which is what the question asked, and one on to meet with, which came up in comments to another answer. You will note in the reference citation that speak is from the OED2 in 1989, whereas talk is from the OED3 in 2001. Only the latter is called out as now being chiefly North American in contemporary usage, and even there many illustrations from British literature of an era slightly earlier than our own are given.
to speak with / to ━━
The OED makes does not call out either of the two phrasal verbs “to speak with ━━” or “to speak to ━━” as being more commonly seen in Britain vs elsewhere.
Here is the first sense for “to speak with ━━ ”:
So if it indeed true that “speaking with someone” is now perceived to be somehow un-British, this is not reflected in the OED as of 1989, nor in the English literature citated.
The phrasal verb “to speak to ━━” has several senses.
Here is the proper citation:
to meet with
In contrast, the OED does say that for the second of its senses, usage of “to meet with” has apparently recently waned in Britain, although it was once common enough. I’ll omit the citations for all but the second sense, as that is the relevant sense for the meaning under discussion.
The requisite source citation is:
As I mentioned in comments, for me, there can be a difference in aspect between meeting someone and meeting with someone. The former can be used for an instance of time, while the latter can be used for a duration.
Therefore you can meet someone at a pub, or meet them for a cup of coffee, or meet them for the first time.
However, when meeting with someone, you can say how long that meeting took place, so I could meet with them for two hours.
Yes, we can see from the OED entries that both "speak to" and "speak with" have been in use in British English for quite a while. However, this is another famous example of a divide between BrE and AmE. In BrE the more common preposition in the twentieth century was "speak TO somebody". Martin Parrott argues that "speak to" is standard (British) English, whereas "speak with somebody" has been stigmatised as a lower-class variant. Under the influence of AmE, the variant "speak with somebody" is considered more acceptable in BrE. Looking forward to ngrams or BNC results! :)
However, even in AmE there is a difference between "speak with" and "speak to". For example, the President of the US speaks to the nation.
As for "meet somebody" and "meet with somebody" - also another famous example - I'll post a quote from Pocket Fowler's:
"meet. There are two uses that deserve attention. 1. It is a transitive verb and so it is possible to meet someone, or simply meet. Idiomatically one meets with a circumstance rather than a person, typically something unpleasant or unwelcome; or one meets with a response or reaction, again often unfavourable (This may not meet with the approval of the parents / Her appointment as chief vet met with sniffiness among some farmers).
Now meet is increasingly used with the link word with (or up with, based on meet up) when people are involved, typically in contexts involving prolonged discussion or dealings (the people who will be meeting with the Secretary of State / A few weeks later I met up with them again). This is not entirely a new use: it is recorded in the OED from the 17c in the sense of casual or accidental encounter, including this quotation of 1740 from David Hume: ‘Tis…rare to meet with persons, who can pardon another any opposition he makes to their interest’. It is true to say however that meet with has become fashionable again in contexts of intentional or arranged meeting under the influence of AmE, and tends to be used in contexts where plain meet would be more usual" [emphasis mine - Alex B.]
"meet" Pocket Fowler's Modern English Usage. Ed. Robert Allen. Oxford University Press, 2008. Oxford Reference Online.