When does the laugh track kick in?
Based on your observation, that spectrum of the signal is sufficiently distinguishable, you can use this as a feature to classify laughter from speech. 

There are many ways you can look at the problem. 

Approach #1

In once case, you can just look at vector of the MFCC. and apply this to any classifier. Since you have many co-efficient in frequency domain, you might want to look at Cascade Classifiers structure with boosting algorithms such as Adaboost based on this, you can compare between speech class vs. laugh class.

Approach #2

You realise that you speech is essentially a time varying signal. So one of the effective ways to do it is to look at time variation of the signal itself. For this, you can split signals in batches of samples, and look at the spectrum for that time. Now, you may realise that laugh might have more repetitive pattern for a stipulated duration where as speech inherently posses more information and hence the spectrum variation would be rather bigger. You can apply this to HMM type of model to see if you consistently remain within the same state for some frequency spectrum or you continuously keep changing. Here, even if occasionally the spectrum of speech resembles to that of laughter it will be more time changing. 

Approach #3

Force to apply LPC/CELP type of coding on the signal and observe the residue. CELP Coding makes a very accurate model of speech production. 

From the reference here: THEORY OF CELP CODING 


  The redundancies in the speech signal are almost removed after the
  short term prediction and long term prediction of the speech signal
  and the residual has very little correlation left in it. Then an
  excitation is searched which synthesizes the speech and the codebook
  index and gain are searched from the fixed codebook. The optimum
  codebook index selection criterion is based on MMSE between the
  locally synthesized speech and the original speech signal.


To put it simply, after all the speech which is predicted from analyser is removed - what is left is the residue which is transmitted to recreate exact waveform. 

How does that help with your problem? Basically, if you apply CELP coding, speech in the signal is mostly removed, what remains is residue. In case of laughter a majority of the signal might be retained because CELP will fail to predict such a signal with vocal tract modelling, where as individual speech will have very little residue. You can also analyse this residue back in frequency domain, to see if it is laughter or speech.