What is the difference
Very often the terms are used interchangeably but not necessarily correctly. The difference is very slight, and essentially boils down to specificity.
Post-and-Beam construction refers to all types of construction where a limited number of posts(vertical members) support beams (horizontal members); as opposed to stud construction where repetitive vertical members are used (Platform framing is in common use today, but balloon framing would also fall under the category of stud construction). Post-and-beam would include both heavy timber and lighter framing, pole-buildings are technically post-and-beam. But most commonly the use of heavy timbers is what comes to mind when post-and-beam is mentioned. The important factor of differentiation is how the connections between post and beam is made. The main divide is whether metal fasteners or wood joinery is used to make the connections. General consensus is that if the connections use metal fasteners and/or hardware then it is not a timber frame but post-and-beam.
What makes it a timber frame
A common definition is that timber frames make connections with a "mortise and tenon" but there are other types of joinery that are commonly used such as laps, birds-mouths, spline joints, and scarf joints to name a few. Sometimes a definitions states that the connections are secured with pegs but they can also be secured by wedges, by interlocking geometry, or not secured at all; many traditional frames had "loose" or unsecured braces.
What is common through all is joinery.
Some more Ambiguity
Historically some timber frames did employ some metal connections but on very limited basis and when there was structural need. For example older frames occasionally use nails to hold down common rafters, straps and bolts reinforced . Purists would argue that these would fall under the category of strictly post-and-beam but since the fastener was used to enhance or reinforce a wooden joint many would say these too are timber frames. Today there are similar "hybrids" that use concealed fasteners, either out of convenience or structural need, and it maybe impossible to tell unless the frame is disassembled.
Another element sometimes seen is that timber frames utilize rectangular timber. But there are some techniques that utilize round, half round, or live-edge timbers; often seen in cruck framing. When logs are stacked horizontally, one atop the other, we are certainly talking about log-building techniques (note that log building uses "joints" as well) which are entirely distinct from post and beam construction. But when the logs are used in a truss or used vertically as a post, it enters the realm of post-and-beam. When connected with joinery timber framing techniques would be the way to go and so we would refine it further.
Another aspect of the timber framing definition is the use of heavy timber. A timber is technically defined as have a cross-section of 5" x 5" or more. Braces often measure 3" x 6" and "nailers" or door and window headers often measure 4" x 4". These scantlings may not be technically be "timbers" but when joinery is involved we consider them as part of the timber frame.
As you can see the heart of the matter is joinery
In short ……
All timber frames are post-and-beam, but not all post-and-beam structures are timber frames.
So if you are talking about heavy timber structures "post-and-beam" would suffice but if you are talking about joinery you are talking about timber framing.