Why can't two series-connected diodes act as a BJT?
Many people think that the answer to this question is related to the width of Base region in BJT transistors - it is incorrect. The answer got quite long. You can read starting from "Tricky Question" section if you want the bottom line.

I believe that you were led to ask this question due to something like this picture:



This is a standard practice of teaching the basics of BJT, but it can confuse someone not familiar with semiconductors theory in details.

In order to answer your question at an acceptable level, I need to assume that you're familiar with the principles of operation of PN diode. This reference contains a detailed discussion of PN junctions. 

The answer concerns NPN transistor, but it also applies to PNP transistors after appropriate change of polarities.



NPN in forward-active mode of operation:

The most "useful" mode of operation of BJT transistor is called "forward-active": 

 

NPN is in forward-active mode when:


Base-Emitter junction is forward biased (usually at \$V_{BE}\approx 0.6V\$)
Base-Collector junction is reverse biased (\$V_{CB}&gt;0\$)


Due to Base-Emitter junction being forward biased, there is an injection of electrons from the Emitter to the Base (\$I_{E_n}\$ in the image above), and the simultaneous injection of holes from the Base to the Emitter(\$I_{B1}=I_{E_p}\$ in the image above). Emitter region (\$n^{++}\$) is much more heavily doped than the Base region (\$p\$), therefore the current due to electrons injected into Base is much higher than the current due to holes injected into Emitter. 

Note that the holes injected into Emitter are supplied from Base electrode (Base current), whereas the electrons injected into the Base are supplied from Emitter electrode (Emitter current). The ratio between these currents is what makes BJT a current amplifying device - small current at Base terminal can cause a much higher current at Emitter terminal. The conventional current amplification is defined as Collector-to-Base currents ratio, but it is the ratio between the above currents which makes any current amplification possible.

Due to injection of huge amount of electrons from Emitter, electrons tend to diffuse through Base to Base-Collector reverse biased junction. Once an electron reaches there, it is swept across the Collector-Base depletion region and is injected into Collector thus contributing to Collector's current (\$I_C\$ in the image above).

Now, if all these electrons injected from Emitter could diffuse to the reverse biased Base-Collector junction without being subject to other effects - there were no importance at all to the width of the Base region. However, there is recombination going on in the Base. 

In recombination process the injected electrons meet holes and "neutralize" each other. The injected electron is "lost" in this process and will not contribute to the current at Collector terminal. But wait, charge conservation requires that the hole which recombined with the injected electron will be supplied from somewhere, right? It turns out that the recombining holes are also supplied from the Base terminal (\$I_{B2}\$ in the image above) thus increasing Base's current and decreasing the Emitter-to-Base currents ratio (which represents transistor's current gain, remember?).

The above means that the more electrons recombine during diffusion through Base region, the lower the current gain of the transistor. It is up to manufacturer to minimize the recombination in order to provide a functional transistor. 

There are many factors that affect recombination rates, but one of the most important ones is Base's width. It is evident that the wider the Base, the more time it will take to injected electron to diffuse through the Base, the higher the chance that it will meet a hole and recombine. Manufacturers tend to make BJTs with very short Base's.

So, why can't two PN diodes back to back function as a single NPN:

The above discussion explained why Base must be short. PN diodes (usually) don't have this short regions, therefore the recombination rate will be very high and the current gain will be approximately unity. What does this mean? It means that the current at "Emitter" terminal will be equal to the current at "Base" terminal, and the current at "Collector" will be zero:







simulate this circuit &ndash; Schematic created using CircuitLab

The diodes are functioning as standalone devices, not a single BJT!

Tricky question:

To various degrees of accuracy, many people can answer your initial question as I did. However, the more interesting question is this: if we make \$p\$ sides of both diodes very short, such that the sum of their widths will be no wider than Base region of NPN transistor, will the diodes function as a transistor?

This question is more difficult to answer because the straightforward answer of "no, the Base of BJT is very short" is not applicable anymore.

It turns out that this approach will not make two diodes similar in behavior to a single NPN transistor. The reason is that at metal contact of the diode, where metal and semiconductor are in touch, all the excessive electrons "recombine" with the "holes" supplied by the contact. It is not the usual recombination as metals don't have holes, but the fine distinction is not that important - once the electrons enter metal, no transistor functionality can be achieved.

The alternative way of comprehending the above point is to realize that Collector-Base diode is reverse biased, but still conducting high current. This mode of operation can not be achieved with standalone PN diodes which conduct a negligible currents under reverse bias. The reason for this restriction is the same - excess electrons from the P side of the forward biased diode can not be swept to the P side of the reverse biased diode through the metal wire in "BJT like diode configuration". Instead, they are swept to the power supply providing a voltage bias to the common terminal of the diodes. 

There was a follow-up question which asked to provide a more rigorous reasoning for the above two paragraphs. The answer concerns metal-semiconductor interfaces and can be found here.

What the above means is that the discussion of the width of Base region is related to discussion of effectiveness of BJT transistors, and is completely irrelevant to the discussion of two back-to-back PN diodes as a substitute for a BJT.

Summary:

Two back-to-back PN diodes can't function as a single BJT because transistor's functionality requires semiconductor only Base region. Once a metal introduced in this path (which is what two back-to-back diodes represent) , no BJT functionality is possible.