Difference between "lots of" and "a lot of"

I think informally, they have the same meaning.

However, the phrase "a lot" refers to an indeterminate unit of measure (which may in certain contexts actually be determinate, e.g. land measurements).

So, technically, "lots of" should be more than "a lot of", since the latter refers to a single unit of measure, and the former refers to multiple units of the same measure (if used in the same context).

Both phrases are very informal; however, there is a denotation for "lot" that indicates "a group" e.g. "a job lot" or, at an auction, one "unit" that is being bid upon.

This is not a very common definition outside of certain markets; dictionary.com gives it as "11. a distinct portion or parcel of anything, as of merchandise: The furniture was to be auctioned off in 20 lots."

Thus, it is entirely possible that "a lot of apples" could refer to a specific grouping of apples being sold or auctioned; if one purchases several of these, then one might refer entirely correctly to "lots of apples."

To avoid confusion, it may be better to use "many" and the verb "to have," e.g. "I have many apples."

You will almost certainly be understood with either of your phrases, but as they are colloquial they may not translate as precisely to non-native speakers as you may wish.

There is no difference. They have the same meaning.

The NOAD I had on my Mac Mini reported the following:

The expressions a lot of and lots of are used before nouns to mean a large number or amount of. In common with other words denoting quantities, lot itself does not normally function as a head noun, meaning that it does not itself determine whether the following verb is singular or plural. Thus, although lot is singular in a lot of people, the verb that follows is not singular. In this case, the word people acts as the head noun and, being plural, ensures that the following verb is also plural: a lot of people were assembled (not a lot of people was assembled).
A lot of and lots of are very common in speech and writing, but they still have a distinctly informal feel and are generally not considered acceptable for formal English, where alternatives such as many or a large number are used instead.
Written as one word, alot is incorrect, although not uncommon.

See also the notes given from the Oxford Living Dictionaries.