Fore one thing the sentance:
It looks like that it's gonna rain today.
is incorrect. One should not use "lile that" in such a construction, and "gonna" is rather informal particularly for writing. One might say:
It looks like it's going to rain today.
It looks like it will rain today.
It looks as if it's going to rain today.
all with much the same meaning.
The phrase "looks like rain" (or "snow" or "a storm" or some other weather condition) is a very common phrase indicating a prediction of rain in the near future, derived from observing the sky and the current weather. From this by metaphor such phrases as "it looks like trouble" are used, in that case a prediction of trouble soon to come.
But with the more general form "It seems that it is going to X" one cannot always say 'It looks like X' in a natural way. This really only works when X is both a noun and a verb, or else when one changes the verb form "to X" into a related noun or noun phrase. Also, the "It is going to X" form implies that X is a natural or general occurrence, for which no cause needs to be specified. One would not say "It seems that it is going to walk" because things in general do not walk, some specific person or animal walks. Only verbs which describe a widespread activity, like a weather condition, fit this construction.