Question: Does having a higher number of gears/speed means the bike will be faster than one with less number of gears/speed, all else being equal?
Short answer: Not necessary (surprise surprise)!
First of all, you need to understand that any single gear is manifested in terms of the number of teeth that a single cog has. In the picture below, the cassette has 6 cogs (including the black one)
A cassette is therefore made up of X number of cogs and may be specified as 11~32T, for example. 11~32T simply means that the smallest cog has 11 teeth and the biggest cog has 32 teeth. However that 2 numbers alone does not tell you how many speed it has, so you need something else that specifies the number of cogs (equivalent to the number of speeds).
For example the number of cogs can be specified as follows:
where each number specifies the number of teeth that each cog has. Yes you are right to say that if there are 10 numbers, then it means there are 10 cogs within that cassette. Compared to the one below where there are 9 numbers:
Although there is only 9 numbers (9 cogs), this particular cassette can also be listed as 11~32T (reference to the first and last number)
The top speed is characterized by the smallest cog (11T) and the slowest speed is characterized by the biggest cog (32T). As you can see, the top and slowest speed are the same for both cassette although both cassettes have different number of cogs in between.
So which one is better? A 9-speed cassette or a 10-speed cassette?
Answer: Again, it depends! (Arrgghh!!!)
The more cogs the cassette has, the smoother the gear change will be. In other words, it will be less abrupt to change speed between adjacent cogs for a cassette that has more cogs. If there are less cogs, it means the change in speed would be more abrupt. It does help in some way to have more cogs although the top and slowest speed remains the same.
However, having more speeds requires a corresponding upgrade for the shifter, derailleur and the bike chain. You should ascertain the total cost of ownership for upgrading/downgrading between the 2, or if you are paying much more outright for a 10-speed cassette vs a 9-speed cassette.
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