Is it better to play $\$1$ on $10$ lottery draws or $\$10$ on one lottery draw?

Your expected gains (or rather losses) are the same for both methods. However, if you get tickets for separate draws, there is an ever so tiny chance that you will win more than once, and correspondingly the chance that you will win (at least once) will be an ever so tiny bit smaller.

As an extreme example of this phenomenon, replace $10$ by the total number of tickets in one draw. Then taking them all in the same lottery ensures a win in that lottery, but taking them in all different lotteries does not ensure any win, but might lead to multiple wins.

Suppose there are only ten tickets - if you buy them all in one draw you have to win. But if you buy in successive draws you can lose every time.

If there is more than one winning ticket in the first case, you could end up winning twice and dividing the pool between your two winning tickets.

In the second case there is a possibility of winning in multiple draws.

So if you are interested in the maximum return for your stake you will need to factor in the value of the win(s) in each case.

In the first case - buying to tickets in a draw with a single win, the tickets represent mutually exclusive events (probabilities add).

In the second case the outcomes are (in the absence of other information) independent - and the easiest way of calculating the probability of winning at least once is first to calculate the probability of losing each time, and multiply using the rule for independent events. Then you should be able to see how to finish this off.

The question is poorly formed, and feels more like game theory than math, per se. By any purely objective standard, the two opportunities provide the same expected outcome. However they have different risk profiles, and so either might be preferable depending on the reward function of the person asking.

If you are optimizing to Maximin, for example, it's better to play ten smaller hands (or not to play at all).