What are the differences in the types of physics, parameterizations, and other properties that they use?
The major differences between weather and climate models are many. At their core lie the same set of primitive equations, but from here there are many differences.
A weather model only (skillfully) predicts about 10 days into the future, while a climate model integrates forward in time for hundreds of years. The main difference here is that in a weather model, we care about the when and where of a storm or front. In a climate model, you get weather, but you don't really care too much where or exactly when the weather is as you are looking for a long term means (e.g. a weather model cares where a hurricane is, and when/where it will impact land, whereas a climate model may only care the average number of hurricanes per year and not about where the details of those storms).
Spatial / temporal resolution
Because climate models run for much longer into the future than a weather model, it will have more integration timesteps for the same temporal scale. This is relaxed by increasing the model timestep, but for numerical stability reasons the higher your timestep the more coarse your spatial resolution must be. As a result, in general, climate models run at larger temporal and spatial scales than weather models. The coarser resolution may force more parameterizations in the climate models. For example a 3 km weather model may explicitly resolve convection, whereas a 30 km climate model will certainly not and need to parameterize convection.
Data Assimilation (DA)
Weather and climate models vary in their use of DA. The biggest difference is how DA is used to "spin up" the model to the initialization time. For weather models, if DA is used at all, you may only have a few DA steps spaced a few hours apart before the model starts integrating into the future. For a climate model the DA period may be 100 years long before the present time is reached and forecasting begun. However:
This incorporation [of DA into climate models] occurs at a number of stages of the model development, including parametrization of sub-grid scale effects and model tuning. The process is not, however, done systematically and current practice is not thought of as "data assimilation." There seems to be a growing realization that DA will have a significant role to play in future climate model development. This is, in part, driven by the need to quantify uncertainty in the model predictions. Nevertheless, there is not a consensus as to how DA should be used in these large-scale climate models. (source: http://www.samsi.info/working-groups/data-assimilation-ipcc-level-models-climate-uq )
Weather models may represent the ocean as a parameterized surface flux (of momentum, moisture, etc) or perhaps handle it through data assimilation. Climate models typically couple the atmosphere model to an ocean model and simulate the ocean as well. The climate models in actuality are typically suites of models that all communicate with each other. You may have a model for atmosphere, one for soil, one for ocean, one for vegetation, one for chemistry, etc. A weather model may have these features, but typically as parameterizations.
Weather models vary from global models to very localized regional models, which can in some cases be very idealized. Climate models tend to be global. This doesn't change the physics involved, but can influence the specific forms of the equations. A global model will solve in spherical coordinates and many use spectral methods. Regional weather models will use Cartesian coordinates and may make other assumptions that simplify the physics for the specific purpose the model (e.g. a storm scale idealized weather model may neglect Coriolis).
This answer is not complete, but it is a start.
One of the most significant differences is:
Weather models use measurements, whereas climate models do not
Put another way: a weather model is an initial value problem. The initial values that go in are of essential importance for the result to be correct.
A climate model solves what is primarily a boundary value problem. The initial values should not matter. In fact, climate models are "spun up", that means, to determine the climate 2000–2100, the model run may start in 1950 to get rid of initial values. Then, the relevant information is not the weather or March 23, 2063; but the statistics of the weather (mean, standard deviation, etc.), over 2060–2090 (for example).
So if a climate model runs 1900–2100, it is not expected to reproduce the climate for a particular year. This is commonly misunderstood, and might be taken by climate skepticists as showing that climate models don't work (for example, “they don't reproduce the recent lack of significant atmospheric warming!”, not realising that this doesn't matter). Different are analysis or reanalysis (weather) models, that do use measurements, and therefore do represent the weather in a particular year accurately.
As stated before, this is only a partial answer. There are many more important differences, but this is one of the most important ones.