Difference between series with drift and series with trend A series with drift can be modeled as $y_t = c + \phi y_{t-1} + \varepsilon_t$
where $c$ is the drift (constant), and $\phi=1$.  
A series with trend can be modeled as $y_t = c + \delta t + \phi y_{t-1} + \varepsilon_t$ where $c$ is the drift (constant), $\delta t$ is the deterministic time trend and $\phi=1$.
Both series are $I(1)$ and I think both exhibit an increasing behavior. 
If I have a new series that exhibits increasing behavior, how do I know this series is a series with drift or with trend?
Can I do two ADF tests:


*

*ADF test 1: Null hypothesis is the series is $I(1)$ with drift 

*ADF test 2: Null hypothesis is the series is $I(1)$ with trend


But what if the null hypothesis for both tests is not rejected?
 A: 
If I have a new series that exhibit an increasing behavior, how do I know this series is a series with drift or with trend?

You may get some graphical clue about whether an intercept or a deterministic 
trend should be considered. Be aware that the drift term in your equation with $\phi=1$ generates a deterministic linear trend in the observed series, while a deterministic trend turns into an exponential pattern in $y_t$.
To see what I mean, you could simulate and plot some series with the R software as shown below.
Simulate a random walk:  
n   <- 150
eps <- rnorm(n)
x0  <- rep(0, n)
for(i in seq.int(2, n)){
  x0[i] <- x0[i-1] + eps[i]
}
plot(ts(x0))

Simulate a random walk with drift:
drift <- 2
x1    <- rep(0, n)
for(i in seq.int(2, n)){
  x1[i] <- drift + x1[i-1] + eps[i]
}
plot(ts(x1))

Simulate a random walk with a deterministic trend:
trend <- seq_len(n)
x2    <- rep(0, n)
for(i in seq.int(2, n)){
  x2[i] <- trend[i] + x2[i-1] + eps[i]
}
plot(ts(x2))


You can also see this analytically. In this document (pp.22), the effect of deterministic terms in a model with seasonal unit roots are obtained. It is written in Spanish but you may simply follow the derivations of each equation, if you need some clarifications about it you may send me an e-mail.

Can I do two ADF tests: ADF test 1. Null hypothesis is the series is I(1) with drift ADF test 2. Null hypothesis is the series is I(1) with trend. But what if for both tests, the null hypothesis is not rejected?

If the null is rejected in both cases then there isn't evidence supporting the 
presence of a unit root. In this case you could test for the significance of the deterministic terms in a stationary autoregressive model or in a model with no autoregressive terms if there is no autocorrelation.
