Is there any functional difference between an odds ratio and hazard ratio? In logistic regression, an odds ratio of 2 means that the event is 2 time more probable given a one-unit increase in the predictor. In Cox regression, a hazard ratio of 2 means the event will occur twice as often at each time point given a one-unit increase in the predictor. Are these not practically the same thing?
What then is the advantage in doing a Cox regression and getting hazard ratios if we can get functionally the same information from the odds ratios of logistic regression?
 A: This is a good question.  But what you are really asking should not be how the statistic is interpreted but what assumptions underlie each of your respective models (hazard or logistic).  A logistic model is a static model which effectively predicts the likelihood of an event occurring at a particular time given observable information.  However, a hazard model or Cox model is a duration model which models survival rates over time.  You might ask a question like "what is the likelihood of a cigarette user surviving to the age of 75 relative to that of a nonuser with your logistic regression" (given that you have information about mortality for a cohort up to 75 years of age).  But if instead you want to take advantage of the fullness of the time dimension of your data then using a hazard model will be more appropriate.
Ultimately though it really comes down to what you want to model.  Do you believe what you are modelling is a one time event? Use logistic. If you believe your event has fixed or proportional chance of occurring each period over an observable time spectrum?  Use a hazard model.
Choosing methods should not be based on how you interpret the statistic.  If this were the case then there would be no difference between OLS, LAD, Tobit, Heckit, IV, 2SLS, or a host of other regression methods. It should instead be based on what form you believe the underlying model you are trying to estimate takes.
A: 
an odds ratio of 2 means that the event is 2 time more probable given a one-unit increase in the predictor

It means the odds would double, which is not the same as the probability doubling.

In Cox regression, a hazard ratio of 2 means the event will occur twice as often at each time point given a one-unit increase in the predictor.

Aside a bit of handwaving, yes - the rate of occurrence doubles. It's like a scaled instantaneous probability.

Are these not practically the same thing?

They're almost the same thing when doubling the odds of the event is almost the same as doubling the hazard of the event. They're not automatically similar, but under some (fairly common) circumstances they may correspond very closely. 
You may want to consider the difference between odds and probability more carefully.
See, for example, the first sentence here, which makes it clear that odds are the ratio of a probability to its complement. So for example, increasing the odds (in favor) from 1 to 2 is the same as probability increasing from $\frac{1}{2}$ to $\frac{2}{3}$. Odds increase faster than probability increases. For very small probabilities, odds-in-favor and probability are very similar, while odds-against become increasingly similar to (in the sense that the ratio will go to 1) reciprocals of probability as probability gets small. An odds ratio is simply the ratio of two sets of odds. Increasing the odds ratio while holding a base odds constant corresponds to increasing the other odds, but may or may not be similar to the relative change in probability.
You may also want to ponder the difference between hazard and probability (see my earlier discussion where I make mention of hand-waving; now we don't gloss over the difference). For example, if a probability is 0.6, you can't double it – but an instantaneous hazard of 0.6 can be doubled to 1.2. They're not the same thing, in the same way that probability density is not probability.
