What probability distribution would be suitable for modelling scores in a basketball match? In football (not American football but what Americans call soccer), it is pretty clear: we may consider the difference of two independent Poisson variables. In basketball, in theory, we could increase the intensity of the Poisson process, however in basketball we also have the shot clock (24 seconds), so it is not really like a more intensive football. I would be most grateful for any ideas.
 A: Without observing actual data from those sports, what you are talking about are essentially just prior beliefs about what the processes might plausibly look like.  Ultimately you will need to test these against data, to see if your model fits the data.  When conjecturing a model based on some independence assumption you will need to look at the data to see if the independence hypothesis actually holds (e.g., check scoring rates over time to see if homogenous Poisson model is okay).  Since modelling of sports scores is a long-established academic field, you also needn't reinvent the wheel - start by reading some of the relevant literature of others who have gone before you.
Football (Soccer): Dixon and Robinson (1998) analyse data from over 4,000 soccer matches and find that there are some obvious deviations from an independent Poisson outcomes.  They find that the rate of goal scoring increases over the course of the match (they hypothesise that this occurs due to tiredness detracting from defense more than from attack) and the scoring rate is dependent on the present score.  To deal with these phenomena they model scores using a non-homogenous Poisson process where the goal rates are able to vary to allow these effects.
Basketball: The main statistical difference in that is likely to occur in modelling basketball is that a single scoring play (defined by there being no game time elapsed between points) can score multiple points.  This means you will need some kind of process that adds a variable number of points during each scoring play.  Gabel and Redner (2012) examine 6,087 games from the NBA and model these using a random walk process.  
If you read the literature on existing models of basketball you will see that they go well beyond independent Poisson random variables.  There are a number of different models in the literature, and the more recent ones are quite complex.  Karlis and Ntzoufrax (2003) give a general examination of bivariate Poisson models in sport, and so this might also be a useful resource.

Dixon, M.J. and Robinson, M.E. (1998) A birth process model for association football matches. The Statistician 47(3), pp. 523-538.
Gabel, A. and Redner, S. (2012) Random walk picture of basketball scoring. Journal of Quantitative Analysis in Sport 8(1), pp. 6-
Karlis, D. and Ntzoufras, I. (2003) Analysis of sports data by using bivariate Poisson models The Statistician 52(3), pp. 381-393.
